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BARYONS FROM LATTICE QCD
A thesis submitted to
Tata Institute of Fundamental Research, Mumbai, India
for the degree of
Doctor of Philosophy
in
Physics
By
M. Padmanath
Department of Theoretical Physics
Tata Institute of Fundamental Research
Mumbai - 400 005, India
May, 2014
Declaration
This thesis is a presentation of my original research work. Wherever contri-
butions of others are involved, every effort is made to indicate this clearly,
with due reference to the literature, and acknowledgement of collaborative
research and discussions.
The work was done under the guidance of Professor Nilmani Mathur, at the
Tata Institute of Fundamental Research, Mumbai.
(M. Padmanath)
In my capacity as the supervisor of the candidate’s thesis, I certify that the
above statements are true to the best of my knowledge.
(Nilmani Mathur)
Acknowledgement
First and foremost, I would like to express my gratitude to my thesis advisor Prof.
Nilmani Mathur for all that I have learned from him during my Ph. D work. It has been
a wonderful experience working with him. I believe I have been able to better myself
over the years through his constant guidance, support and encouragement. I gratefully
acknowledge Prof. Saumen Datta for all the excellent discussions, which helped me learn
many aspects about QCD and lattice, with him. I am equally grateful to Prof. Mike
Peardon for all the help and suggestions and giving me opportunity to spend time at
Trinity College, Dublin. I would also like to thank Prof. Sourendu Gupta, Prof. Robert
Edwards and Dr. Christopher Thomas for all the fruitful discussions that resulted in
many of the contents discussed in this thesis. For making the time spent in the office
really fun and for numerous physics discussions, I am grateful to all my DTP colleagues,
many of whom are elsewhere at present. Special thanks to Dr. Debasish Banerjee, Nikhil
Karthik, Dr. Jyotirmoy Maiti and Dr. Sayantan Sharma for all their help during my
Ph. D. I would like to acknowledge all the help from the DTP office staff : Girish
Ogale, Mohan Shinde, Rajan Pawar and Raju Bathija. Special thanks also are due to
the system administrators Kapil Ghadiali and Ajay Salve for being extremely helpful all
the times. Thanks are also due to the ILGTI (Indian Lattice Gauge Theory Initiative)
for making available the computing resources of Blue gene, CRAY and three clusters of
the Department of Theoretical Physics - GAGGLE, HIVE and BROOD, on which most
of the computations reported in the thesis were done. I would like to acknowledge the
support during the course of my Ph. D. from the Council of Scientific and Industrial
Research(CSIR), Government of India through the SPM fellowship and the support from
the Trinity College Dublin Indian Research Collaboration Initiative.
I have to thank a lot of friends whose help and support made my life in TIFR enjoyable.
Specially I would like to appreciate and acknowledge the constant company from Carina,
Debjyoti, Mathimalar, Rajeev, Rishad, Sasidevan, Shyama and Thomas among others
whom I have not named explicitly : my stay at TIFR would not have been so much
enjoyable without their presence. I would also like to thank all my friends in the Malayali
community in TIFR for a wonderful time I had with them. It was a joy to have your
company and because of you, I didn’t feel like missing my home. Finally, I would like
to thank my parents, Madanagopalan and Kalavathy, and my brother, Premnath, for
always being extremely supportive and for all their love and care they have for me.
Collaborators
This thesis is based on the work done in collaboration with Subhasish Basak, Saumen
Datta, Robert G. Edwards, Sourendu Gupta, Andrew Lytle, Jyotirmoy Maiti, Pushan
Majumdar, Nilmani Mathur and Mike Peardon. In particular, Chapter 3 and 4 was done
in collaboration with Robert G. Edwards, Nilmani Mathur and Mike Peardon (2013-);
Chapter 5 with Subhasish Basak, Saumen Datta, Andrew Lytle, Pushan Majumdar and
Nilmani Mathur (2012-); Chapter 5 with Saumen Datta, Sourendu Gupta, Jyotirmoy
Maiti and Nilmani Mathur (2013).
To my family
Contents
Synopsis iv
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
0.2 Lattice methodology in excited state spectroscopy . . . . . . . . . . . . . x
0.2.1 Anisotropic lattice . . . . . . . . . . . . . . . . . . . . . . . . . . xi
0.2.2 Construction of baryon operators . . . . . . . . . . . . . . . . . . xii
0.3 RESULTS : Excited state spectroscopy of charm baryons . . . . . . . . . xviii
0.3.1 Triply charm baryons . . . . . . . . . . . . . . . . . . . . . . . . . xviii
0.3.2 Doubly charm baryons . . . . . . . . . . . . . . . . . . . . . . . . xxi
0.3.3 Singly charm baryons . . . . . . . . . . . . . . . . . . . . . . . . . xxii
0.4 Mixed action approach : Charm and strange hadron spectroscopy . . . . xxv
0.4.1 Energy splittings in charmonia and charm-strange mesons . . . . xxvi
0.5 Baryon screening masses at finite temperature . . . . . . . . . . . . . . . xxvii
0.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . xxix
Publications xxxviii
Outline of the thesis xxxix
1 Introduction 1
1.1 Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Heavy hadrons in QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Potential model calculations . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Lattice spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Excited state spectroscopy . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Spectroscopy with chiral fermions . . . . . . . . . . . . . . . . . . 10
1.3 Hadrons at finite temperature . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
i
Contents ii
2 Quantum ChromoDynamics on lattice 14
2.1 QCD in the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Lattice formulation of QCD . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Gauge action on lattice . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 Fermion action on lattice . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Numerical calculations using lattice QCD . . . . . . . . . . . . . . . . . . 23
2.3.1 Simulations using lattice QCD . . . . . . . . . . . . . . . . . . . . 23
2.3.2 Spectroscopy on the lattice . . . . . . . . . . . . . . . . . . . . . . 26
2.3.3 Quark propagators . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Excited state spectroscopy 29
3.1 Conventional spectroscopy from lattice . . . . . . . . . . . . . . . . . . . 29
3.2 Challenges in excited state heavy hadron spectroscopy . . . . . . . . . . 31
3.3 Anisotropic lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Construction of baryon operators . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Continuum baryon interpolating operators . . . . . . . . . . . . . 35
3.4.2 General symmetry combinations . . . . . . . . . . . . . . . . . . . 36
3.4.3 Flavor symmetry structures . . . . . . . . . . . . . . . . . . . . . 37
3.4.4 Dirac spin symmetries . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.5 Spatial projection operator symmetries . . . . . . . . . . . . . . . 41
3.4.6 Subduction into lattice irreps . . . . . . . . . . . . . . . . . . . . 44
3.4.7 Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Variational fitting technique . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Rotational symmetry and cross correlations . . . . . . . . . . . . . . . . 52
3.7 Spin identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Charm baryon spectrum 64
4.1 Triply charm baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.1 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.2 Investigating lattice artifacts . . . . . . . . . . . . . . . . . . . . . 68
4.1.3 Valence quark mass dependence of energy splittings . . . . . . . . 69
4.2 Doubly charm baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.1 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Contents iii
4.2.2 Investigating lattice artifacts . . . . . . . . . . . . . . . . . . . . . 81
4.2.3 Energy splittings in doubly charm baryons . . . . . . . . . . . . . 83
4.3 Singly charm baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.1 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.2 Σc and Ωc baryon . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3.3 Λc baryon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3.4 Ξc baryon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.5 Energy splittings : A preliminary study . . . . . . . . . . . . . . . 94
4.4 Caveats in excited state spectroscopy . . . . . . . . . . . . . . . . . . . . 96
4.5 Summary, conclusions and future prospects . . . . . . . . . . . . . . . . . 99
5 Spectroscopy using chiral fermions 102
5.1 Mixed action formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 Numerical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3 Discretization errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3.1 Finite momentum meson correlators . . . . . . . . . . . . . . . . . 106
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4.1 Hyperfine splitting in 1S charmonia . . . . . . . . . . . . . . . . . 109
5.4.2 Energy splittings in charmonia and charmed-strange mesons . . . 111
5.4.3 Charmed baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.5 Summary, conclusions and future prospects . . . . . . . . . . . . . . . . . 115
6 Baryons at finite temperature 116
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2 Runs and measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3 The meson sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 The baryon sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5 Hadrons above Tc and free field theory . . . . . . . . . . . . . . . . . . . 130
6.6 Summary, conclusions and future prospects . . . . . . . . . . . . . . . . . 131
6.A Exceptional configurations . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.B Wall sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7 Summary and conclusions 140
7.1 Excited state charm baryon spectroscopy . . . . . . . . . . . . . . . . . . 140
7.2 Charm hadron spectroscopy using overlap fermions . . . . . . . . . . . . 142
7.3 Nucleons at finite temperature . . . . . . . . . . . . . . . . . . . . . . . . 142
Synopsis
Abstract
In this synopsis, I report on non-perturbative studies of various aspects of baryon physics us-
ing lattice QCD. The first part of the thesis consists of spectroscopy of baryons with one or
more charm quarks using two complementary non-perturbative approaches. The first approach
enables us to extract highly excited states, which are very difficult to extract using the previ-
ously existing lattice QCD techniques, of charm baryons and the second allows us to extract
the ground states of various charm and strange hadrons with improved control over cut-off
effects. The second part of the synopsis deals with a study of baryon screening masses at finite
temperature.
In the work on excited state charm baryon spectroscopy, which constitutes the major part
of this thesis, we use anisotropic lattices having Nf = 2 + 1 dynamical quark fields with very
fine temporal lattice spacing, which is crucial for minimizing the discretization errors. After
constructing the correlation functions with a large basis of interpolating operators employing
a recently developed technique, called distillation, we extract the highly excited states using a
variational fitting method. The spectra of excitations of charm baryons thus computed have
baryonic states with well-defined total spin up to 72 for both the parities and the low lying
states closely resemble the pattern expected from non-relativistic potential models. We also
study various energy splittings between the extracted states, which can provide insights on the
interactions within heavy baryons.
Complementary to the above work, we perform lattice calculations of charm hadrons adopt-
ing a mixed action approach with the overlap action for the valence quarks on a background
of 2+1+1 flavors highly improved staggered quarks. We calculate the low lying hadron spectra
as well as energy splittings at various lattice spacings with improved control over cut-off effects
and compare with results from the above work.
In the second part, we perform a finite temperature study of screening masses of hadrons,
particularly for baryons, where we analyze the temperature dependence of the hadronic screen-
ing correlators above and immediately below the deconfinement transition temperature (Tc).
iv
Synopsis v
Above Tc, both mesonic and baryonic screening correlators show clear evidence for weakly in-
teracting quarks, while below Tc, unlike mesonic channels, baryonic screening correlators show
precursor effects for chiral symmetry restoration.
Synopsis vi
0.1 Introduction
Most of the mass of the visible universe - about 99% - constitute of baryons. Though
the recently discovered Higgs boson [1] is believed to provide mass to the fundamental
particles like leptons and quarks, it does not explain the total mass of the baryons that
are made of these quarks. Baryons are relatively simple systems in which quintessentially
confining character of multiple quarks is manifest and are sufficiently complex to shed
light on physics hidden from us in the mesons. Understanding the baryon spectra is
thus expected to answer a large series of questions on baryons and the interactions that
exist within them. Particularly, with the recent excitement in heavy hadron physics and
fore-seeing the observations from large statistical samples that will be collected in many
future and ongoing experiments, a detailed understanding of the heavy baryon spectra
is expected to provide insights into various aspects of the strong force that cannot be
probed with light baryons.
The physical properties of baryons are believed to be governed by the theory of
Quantum ChromoDynamics (QCD), an asymptotically free local non-Abelian quantum
gauge field theory. As one of four recognized fundamental forces of nature, the theory
of QCD, also known as the strong force, is one of the most important areas of research
in modern particle physics. In high energy domain, where the coupling constant is small
enough for pursuing a perturbative study, analytical calculations based on QCD have
been proved to be highly accurate in explaining numerous experimental data, such as in
deep inelastic scattering. However, at low energies, the coupling constant becomes very
large, rendering perturbation theory in the coupling inapplicable.
A comprehensive understanding of the theory of QCD in the strong coupling regime
implies a rigorous determination and understanding of its bound states. If QCD is the
correct theory, then one should be able to reproduce the physical hadron spectra from a
high precision first principles non-perturbative calculation. Much like the role of atomic
spectroscopy in the development of quantum mechanics, hadron spectroscopy, both the-
oretically and experimentally, has played a crucial role in understanding the nature of
the fundamental strong force and its degrees of freedom. The rich spectra of physical
hadrons for light quarks provided us with a framework for constructing the theory of
strong interaction starting from eightfold way to quarks-partons as the fundamental de-
grees of freedom. The discovery of the heavier hadron, J/ψ meson, and the subsequent
discoveries of other charmonia put this framework on solid footing. The discovery of J/ψ
meson triggered so huge scientific interest that it was termed as the November revolution.
Synopsis vii
From a mere mathematical construct, the concept of quarks became reality. The charm
quark being heavier in mass, the approximation that the charm quark and anti-quark are
non-relativistic were found to be good for many predictions. Studies on various energy
splittings provided crucial information on heavy quark-antiquark potential, hyperfine and
spin-orbit interactions, etc. The ground state charmonia, J/Ψ and ηc, gave an excellent
example of a kind of hydrogen atom of strong interactions, which revealed the quark layer
of substructure for hadrons. Potential models, consisting of non-relativistic quark kinetic
energy, a central confining potential, and spin-dependent interaction terms were found
to be very successful in explaining various observed states. Recently, a tower of heavy
hadron states, including the X’s, Y ’s and Z’s, have been discovered with unusual proper-
ties [2]. These discoveries have rejuvenated the heavy hadron spectroscopy tremendously
and offer a bright promise in detailed understanding of various interesting aspects of the
theory of strong interaction.
While the heavy quarkonia and other heavy flavored mesons have been studied quite
extensively, the heavy baryon physics has received substantially less attention, even
though they can provide similar insight in understanding the strong interaction. Just
as the quark-antiquark interactions are examined in charmonia, these studies can probe
the interactions between multiple heavy quarks and heavy quarks with one or more
lighter quarks. Various spin dependent splittings between baryons can provide informa-
tion to constrain spin dependent potential terms which are crucial to build successful
Non-Relativistic QCD (NRQCD), pNRQCD and similar models. Experimentally only a
handful of charm baryons have been discovered and a reliable determination of the quan-
tum number of most of the observed states has not been made [2]. Only very recently have
a few excited singly charm baryons been observed and the discovery status of the doubly
charm baryons remains unsettled. While the SELEX experiment observed doubly charm
Ξcc(ccu) baryons [3], these have not been confirmed either by BABAR [4] or Belle [5].
Along with the well-established triply flavored ∆(uuu) and strange Ω(sss) baryons, QCD
predicts similar states built from charm quarks, the triply charm baryon, Ωccc. Such a
state is yet to be observed. This could be because of the large energy threshold re-
quired for their production via either resonant or continuum production mechanisms,
their very short lifetime and the very low reconstruction efficiency for the highly excited
heavy hadron resonances, which follows cascade decays into multi-particle final states.
The extremely low production rate hinders the identification of the spin-parity quantum
numbers, even though the partial wave analysis is relatively simpler in heavy hadrons in
comparison with the light hadron spectra. However, it is expected that the large sta-
Synopsis viii
tistical sample collected at the LHCb experiment, the PANDA experiment at the FAIR
facility, Belle II at KEK and BES III may be able to provide some information on charm
baryons.
On the theoretical side, one expects that potential models will be able to describe,
charm baryons [6, 7, 8], in particular the triply charm baryons, to a similar level of
precision as their success in charmonia. The triply charm baryons may provide a new
window for understanding the structure of baryons, as pointed out by Bjorken several
years ago [9]. The spectra of charm baryons have been studied theoretically [6, 7, 8] using
non-relativistic potential models and various improvised versions of that.
QCD, like other quantum field theories, needs to be regularized. Regularization using
a lattice discretization of space-time (known as lattice QCD) has the advantage that
one can use numerical techniques to access the non-perturbative aspects of QCD. A
quantitative description of the spectra of the charm baryons using rigorous ab-initio
computations like lattice QCD are important for a number of reasons. Firstly, all the
estimates of physical states from these calculations will be predictions and thus can
naturally provide crucial inputs to the future experimental discovery. Secondly, lattice
results can provide a guide in identifying unknown quantum numbers (spin-parity) of the
discovered states, based on what one expects from QCD. Moreover, it will be interesting to
compare the low lying spectra of charm baryons computed from a first principles method
to those obtained from potential models which have been very successful for charmonia.
It is expected that more information about interactions between multiple charm quarks
and, charm quark and light quarks can be obtained by computing the excited state
spectra of charm baryons, including in particular the spin-dependent energy splittings.
The success of lattice QCD calculations in light hadron spectroscopy is well docu-
mented, e.g. Ref. [10, 11, 12, 13, 14]. There were quite a few successful heavy hadron
spectroscopic studies, for mesons, e.g. [15, 16, 17, 18, 19, 20, 21], and for baryons, e.g.
[22, 23, 24, 25, 26, 27, 28, 29], but they were limited only to the ground state determi-
nations. In spite of all these motivating results from lattice QCD, estimation of excited
state spectrum remained a major challenge till last 7-8 years. Invention of novel smear-
ing techniques called distillation [30] and the proper utilization of fitting techniques, such
as the generalized eigenvalue method [31, 32], along with the derivative based operator
construction formalism [33, 34], boosted the hadron spectroscopic studies, facilitating the
determination of the excited state spectra [34, 35, 36, 37, 38].
The major part of this synopsis consists of spectroscopy of charm baryons using two
complementary non-perturbative approaches. The first, which constitutes the major part,
Synopsis ix
employs these novel techniques mentioned above, enabling us to extract a tower of spin
identified excited states for all charm baryons. Using an anisotropic lattice with very fine
temporal resolution and a large set of carefully constructed interpolating operators, we
study the ground as well as the excited states of charm baryons with one or more valence
charm quark content for each spin-parity channel up to spin 7/2. We also computed
several energy splittings, for example, hyperfine as well as spin-orbit splittings between
various states. The results on these studies were reported in Ref. [39].
In the above mentioned work, we used the tree level tadpole improved clover action
with no O(mat) and O(mas) errors. We have not addressed other higher order terms, as
the temporal lattice spacing is quite small. We also do not make a quantitative study of
O(m2a2s) errors. However, to study the charm physics, it is essential to have an action
with discretization errors as small as possible. Another caveat in the above study is that
the clover action does not have chiral symmetry at finite lattice spacing, i.e., we have not
addressed the effects of chiral symmetry on observables. Keeping these caveats in mind,
confronting this study with a totally different non-perturbative approach, having better
control over the systematics, will be a good check to validate the smallness of systematic
corrections in the above results.
Hence to complement the above work, we performed another non-perturbative calcu-
lation of the low lying spectra using overlap valence quarks in the background of a large
set of 2+1+1 flavors dynamical configurations generated with the one-loop, tadpole im-
proved Symanzik gauge action and the highly improved staggered quark (HISQ) fermion
action [40, 41]. By adopting such a mixed action approach, one can get advantage of
the chiral symmetry and low quark mass limit of overlap fermions, and the advantage of
having a large set of configurations with small discretization errors as well as small taste
breaking effects. One also gets the advantage of simulating both light, strange as well
as heavy fermions on the same lattice formalism with chiral fermions having no O(a) er-
rors. With the above formulation we have calculated the ground state spectra of various
charm hadrons and have made a comparative study between these two projects along
with other results from literature where available. Agreement between these results from
different approaches gives confidence in our estimates. The results from these studies
were reported in Ref. [27].
In the third work, we performed a finite temperature analysis of hadronic screening
correlators [42], with main focus on the baryon sector. The question is what are the
degrees of freedom that exist in a QCD medium at finite temperature and how they are
modified across the crossover / transition temperature. These can be answered by study-
Synopsis x
ing the static correlation lengths in a QCD medium in thermal equilibrium. In this work,
we focused on the static spatial correlations in the equilibrium thermodynamic system
for probes with mesonic and baryonic quantum numbers. We performed simulations of
pure gauge theory for three different temperatures across the transition temperature and
studied screening correlators for mesonic and nucleonic resonances with clover fermions.
0.2 Lattice methodology in excited state spectroscopy
Lattice QCD is the first principles non-perturbative numerical technique widely used
in determining the QCD observables. Introduced by Kenneth Wilson in 1974 [43], it
regularizes the theory by discretizing the Euclidean space-time and representing QCD
Lagrangian on a finite size lattice. Expectation values of the QCD observables can be
expressed as path integrals which can be directly evaluated with controlled systematics
using Monte Carlo methods.
Lattice computations of hadron masses proceed through the calculations of the Eu-
clidean two point correlation functions, Cji, between creation operators, Oi, at time ti
and annihilation operators, Oj, at time tf .
Cji(tf − ti) = 〈0|Oj(tf )Oi(ti)|0〉 =∑n
Zn∗i Z
nj
2mn
e−mn(tf−ti) (1)
The RHS is the spectral decomposition of such two point functions, where the sum is over
a discrete set of states, n, with energies, mn. Zn = 〈0|Oi|n〉 is the vacuum state matrix
element, also called an overlap factor. In general, the calculation of hadronic masses by
lattice QCD methods takes place by performing non-linear exponential fits to the long
distance tail of these two point correlators, where only the ground state contribution
dominates.
The goal of this work is to calculate the ground as well as the higher excited state
spectra of charm baryons. However, one encounters several problems while extracting the
masses of these states by using the naıve method as mentioned. The first major hurdle
in charm hadron spectroscopy is attributed to the heavyness of the charm quark. One
needs to use a very fine lattice so that the mca << 1 to keep discretization error as small
as possible. Moreover, the physical states comprised of valence charm quarks are quite
heavy and their temporal correlation functions decay increasingly rapidly in the Euclidean
time with increase in the energy of the resonances, while the noise behaves in the same
Synopsis xi
manner as in the ground state. Hence the signal-to-noise ratio in the correlators of these
states exhibits increasingly rapid degradation in Euclidean time with increasing energy.
A remedy of these problems is to use an anisotropic lattice, as discussed in Section 0.2.1.
Another difficulty is in the extraction of multiple excited states from a single correlator
using a reliable non-linear fitting method. One way to get away with this issue is to
use correlation functions for multiple interpolators as well as to use a better analysis
method to extract excited states from them reliably. A careful construction of large basis
of interpolating operators that could overlap strongly with the desired excited states is
crucial for this purpose. In Section 0.2.2, we discuss the construction of a large basis
of interpolating operators and the novel smearing technique that provides an efficient
method to compute the two point correlation functions. In addition to the operator
construction, the extraction of the physical states with correct quantum numbers from
correlation functions of such large basis of operators, relies on a good analysis procedure.
In Section 0.2.2, we discuss the variational fitting method that we have used to extract
the spectrum of baryon states from matrix of correlation functions using such large basis
of operators. A reliable identification of the spin-parity quantum numbers of a state is
highly non-trivial on a finite lattice. However, it is possible to identify these quantum
numbers on lattices with even single lattice spacing using the overlap factors as described
in Section 0.2.2 [32].
0.2.1 Anisotropic lattice
A possible solution for the issues related to discretization error as well as the rapid
decay of the temporal correlations is to increase the temporal resolution by adopting a
finer lattice spacing along the Euclidean time direction while keeping the spatial lattice
constants same, so that one can avoid the computational cost that would come from
reducing the spacing in all directions[44, 45]. In this work, we adopt such an anisotropic
dynamical lattice formulation to extract the highly excited charm baryon spectra.
We used the anisotropic Nf = 2 + 1 flavor dynamical gauge configurations generated
by the Hadron Spectrum Collaboration (HSC) with clover improved Wilson fermions for
both sea and valence quarks. The gauge field sector utilized the tree-level Symanzik-
improved gauge action, while the fermion sector used an anisotropic Shekholeslami-
Wohlert fermion action with tree-level tadpole improvement and three-dimensional stout-
link smearing of gauge fields. More details of the formulation of actions as well as the
techniques used to determine the anisotropy parameters can be found in Refs. [45, 46].
Synopsis xii
Lattice size atm` atms Ncfgs mK/mπ mπ/MeV atmΩ
163 × 128 −0.0840 −0.0743 96 1.39 391 0.2951(22)
Table 1: Properties of the gauge-field ensembles used. Ncfgs is the number of gauge-fieldconfigurations.
The lattice action parameters of the gauge-field ensembles used in this work are given
in Table 3.1. The temporal lattice spacing at was determined using the Ω-baryon mass
measured on these ensembles [46]. This leads to a−1t = 5.67(4) GeV. On this lattices we
find mcat << 1, which assures the use of these lattice to study the charm physics. With
an anisotropy close to 3.5, as = 0.12 fm. This gives a spatial extent of about 1.9 fm,
which would be sufficient to study charm baryons, in particular doubly and triply charm
baryons.
The details of the charm quark action used in this study are given in Ref. [37]. The
action parameters for the charm quark are obtained by ensuring that the mass of the ηc
meson takes its physical value and its dispersion relation at low momentum is relativistic.
It is expected that the effects due to the absence of dynamical charm quark fields in this
calculation will be small, as the disconnected diagrams are OZI suppressed.
0.2.2 Construction of baryon operators
In this section, we briefly discuss the construction of baryon interpolating operators using
up to two derivatives, such that they project up to J = 72
states with both even and odd
parities. The construction has been detailed in [34]. Hence in this synopsis, we discuss
only the basics and the essentially different part from that in Ref. [34], which are the
flavor structures. The details about the construction of the operators used for the study
also provides insight into the results obtained from the numerical studies.
The methodology involves first constructing a basis of baryon interpolating operators
in the continuum with well-defined continuum spin-parity quantum numbers. These
continuum operators are then subduced/reduced to the irreducible representations (irrep)
of the octahedral group on the lattice. The correlation functions, constructed out of
these octahedral operators, are expected to show rotational symmetry breaking artefacts
in these calculations due to the reduced symmetry on a lattice. However, we see an
effective rotational symmetry from the primary spin identification tests and that the
symmetry breaking effects are very small.
Synopsis xiii
Continuum baryon interpolating operators
All hadrons are color singlet objects and thus they form totally anti-symmetric combi-
nations in the color indices of its constituents. Since baryons are fermions made of three
quarks, their interpolating operators excluding the color part should be totally symmetric
combinations of all the quark labels representing flavor, spin and the spatial structure.
The overall flavor-spin-spatial structure of a continuum baryon interpolating operator
with quantum numbers, JP , can be decomposed into a combination as
O[JP ] = [FΣF ⊗ SΣS ⊗DΣD ]JP
, (2)
where F , S and D are flavor, Dirac spin and spatial projection operators respectively and
the subscripts stands for the symmetry in the respective subspaces. For every baryon
operator, we must combine the symmetry projection operators such that the resulting
baryon operator, excluding the color part, is overall symmetric.
Flavor symmetry structures
With only three different flavors simultaneously possible, the flavor wave functions of
the charm baryon operators can be constructed to be members of SU(3)F multiplets as
3⊗3⊗3 = 10S⊕8MS⊕8MA⊕1A, where S, MS, MA and A stands for Symmetric, Mixed-
Symmetric, Mixed-Antisymmetric and Antisymmetric constructions. These represent a
subset of states from the larger group of SU(4)F which has a decomposition based on
symmetry considerations as 4⊗ 4⊗ 4 = 20S ⊕ 20MS ⊕ 20MA⊕ 4A. Table 3.2 contains the
details of the flavor symmetry constructions possible for various charm baryons.
For the spin and spatial parts also, we construct similar symmetry structures.1 For the
spatial symmetry structures we considered operators that include up to two derivatives.
Totally symmetric constructions in flavor-spin-spatial indices from combinations of these
flavor symmetry structures and the symmetry structures in the spin and the spatial parts
forms the desired local and non-local charm baryon interpolating operators [34].
Only a subset of operators that are formed purely from the upper two components
of the Dirac spinor, in Dirac-Pauli representation, appear at a leading order velocity
expansion and hence are referred to as “non-relativistic”. All other operators other than
this subset is referred to as “relativistic”. Using non-relativistic components alone, it
is not possible to construct a negative parity state with a spin higher than 5/2, even
1The details about the spin-spatial symmetry structures are omitted for the sake of space.
Synopsis xiv
20MI Iz S FMS FMA
Λ+c 0 0 0 1√
2(|cud〉MS − |udc〉MS) 1√
2(|cud〉MA − |udc〉MA)
Σ++c 1 +1 0 |uuc〉MS |uuc〉MA
Σ+c 1 0 0 |ucd〉MS |ucd〉MA
Σ0c 1 −1 0 |ddc〉MS |ddc〉MA
Ξ′+c
12 +1
2 −1 |ucs〉MS |ucs〉MA
Ξ′0c
12 −1
2 −1 |dcs〉MS |dcs〉MA
Ξ+c
12 +1
2 −1 1√2(|cus〉MS − |usc〉MS) 1√
2(|cus〉MA − |usc〉MA)
Ξ0c
12 −1
2 −1 1√2(|cds〉MS − |dsc〉MS) 1√
2(|cds〉MA − |dsc〉MA)
Ω0c 0 0 −2 |scs〉MS |scs〉MA
Ξ++cc
12 +1
2 0 |ccu〉MS |ccu〉MA
Ξ+cc
12 −1
2 0 |ccd〉MS |ccd〉MA
Ω+cc 0 0 −1 |ccs〉MS |ccs〉MA
20SI Iz S FS
Σ++c 1 +1 0 |uuc〉S
Σ+c 1 0 0 |ucd〉S
Σ0c 1 −1 0 |ddc〉S
Ξ+c
12 +1
2 −1 |ucs〉SΞ0c
12 −1
2 −1 |dcs〉SΩ0c 0 0 −2 |ssc〉S
Ξ++cc
12 +1
2 0 |ccu〉SΞ+cc
12 −1
2 0 |ccd〉SΩ+cc 0 0 −1 |ccs〉S
Ω++ccc 0 0 0 |ccc〉S
4AI Iz S φA
Λ+c 0 0 0 |udc〉A
Ξ+c
12 +1
2 −1 |ucs〉AΞ0c
12 −1
2 −1 |dcs〉A
Table 2: Flavor symmetry structures for charm baryons. I in the first column stands forthe isospin, Iz in the second column is for the third component of isospin and S in thethird column is for the strangeness.
with operators that include two derivatives. Use of relativistic operators along with non-
relativistic ones enables us to extract states with spin up to 7/2 for both the parities. A
subset of operators constructed using two derivatives and projected onto L = 1 orbital
angular momentum with MS/MA, are proportional to the field strength tensor and are
zero in a theory without gauge fields. These operators are identified as hybrid operators.
Account of the orbital motion classifies the total symmetry in a non-relativistic approach
with 3 flavors to be SU(6)⊗ O(3) super-multiplets with the O(3) describing the orbital
motion. With four component Dirac spinors, the actual construction of the operators
corresponds to SU(12)⊗O(3).
Synopsis xv
Subduction into lattice irreps
Ωccc G1 H G2
SF g u g u g uTotal 20 20 33 33 12 12
Hybrid 4 4 5 5 1 1NR 4 1 8 1 3 0
(b)
Ξcc, Ωcc G1 H G2
(SF +MF ) g u g u g uTotal 55 55 90 90 35 35
Hybrid 12 12 16 16 4 4NR 11 3 19 4 8 1
(a)
Σc, Ωc (SF +MF ) Λc (MF + AF ) Ξc (SF +MF + AF )G1 H G2 G1 H G2 G1 H G2
g u g u g u g u g u g u g u g u g uTotal 55 55 90 90 35 35 53 53 86 86 33 33 116 116 180 180 68 68
Hybrid 12 12 16 16 4 4 12 12 16 16 4 4 24 24 32 32 8 8NR 11 3 19 4 8 1 10 3 17 4 7 1 23 6 37 8 15 2
(c)
Table 3: The number of lattice operators for different charm baryons obtained after sub-duction to various irreps from continuum operators with up to two covariant derivatives.The number of non-relativistic (NR) and hybrid operators for each irreps and for botheven (g) and odd (u) parities are given.
In lattice QCD computations, one considers the theory in a discretized four dimen-
sional Euclidean hyper-cubic box. The hyper-cubic lattice does not possess the full sym-
metry of the continuum. Thus, in a lattice calculation, the states at rest are classified
not according to the continuum spin (J ,Jz), but rather according to the irreps of a cube.
There are eight irreps of the double-valued representations of the octahedral group (Oh),
named as A1, A2, E, T1 and T2 for the integer spins, and G1, G2 and H representing the
half-integer spins. For baryons, there are four two-dimensional irreps corresponding to
G1, G2 (two for each parity) and two four dimensional representations for H. The G1 irrep
contains J = 12, 7
2, 9
2, 11
2, ... states, the G2 irrep contains J = 5
2, 7
2, 11
2, ... states while H irrep
has J = 32, 5
2, 7
2, 9
2, ... states. One needs to examine the degeneracy patterns across the
different Oh irreps to reliably deduce continuum-limit spins J of these states which will
be discussed later. Mathematically this subduction of continuum spin to lattice irreps
can be expressed as,
ΩJnΛ,r =
∑m
SJ,mnΛ,r ΩJm, (3)
Synopsis xvi
where ΩJnΛ,r is the lattice operator representing the rth row of the nth embedding of
the irrep Λ on the continuum spin J , subduced from a continuum spin operator ΩJm.
The SJ,mnΛ,r are called the subduction coefficients. Table 3 shows the total number of
relativistic and non-relativistic operators constructed based on eq. (3.15) for various
charm baryons. It also contains the number of non-relativistic and hybrid operators
constructed for various charm baryons.
The complexity of these operators is evident from this example. A continuum Ωccc
operator with spin and parity of the three quarks as 32
−in the first embedding of mixed-
symmetry Dirac spin constructions and with two derivatives coupled into L = 2 with
a mixed-symmetric spatial part and total spin and parity JP = 72
−, is denoted as(
Ωccc,S ⊗ (Hu,1,M
)⊗D[2]L=2,M
)JP= 72
−
. On the lattice this operator gets subduced once
over G1u, G2u and Hu.
Distillation
Having constructed a large set of operators it is possible to calculate their diagonal and
off-diagonal correlation functions to extract excited states by using variational methods.
However, construction of correlation functions with so many operators, particularly for
the operators with one or more derivatives, is very computationally intensive as it needs
one inversion for each non-local source. The total time needed for the construction of
correlation function could be as intensive as gauge field generations. To avoid this we
use a recently developed novel technique, called the “distillation method” [30]. Using
this method, one can efficiently construct a large number of correlation functions with
multiple operators. Moreover, this method automatically includes a smearing process, a
requirement for suppressing the high frequency modes that do not contribute significantly
to the low energy regime in the spectrum and thus increases the relative contributions
from the low lying states in the correlation functions.
In the distillation method one uses a smearing operator which is a projection operator
constructed by outer product over the low lying eigenmodes of the discretized gauge-
covariant Laplacian, ∇2, as
xy(t) = Vxz(t)V†zy(t) =
Nvecs∑k=1
ξ(k)x (t)ξ(k)†
x (t), (4)
where ξ(k)x are Nvecs number of eigenvectors, sorted by the eigenvalues, of ∇2 evaluated
Synopsis xvii
on the background of spatial gauge fields of the time slice t. The number of these modes,
Nvecs (volume dependent), should be sufficient to sample the required angular structure
of the low lying hadronic states, but is small compared to the number of sites on a time
slice. When Nvecs is the same as the number of sites on a time-slice, then the distillation
operator becomes identity, and the fields acted upon are unsmeared. The computational
cost is highly reduced because inversions need to be performed in a smaller space of
Dirac spin and distillation vectors, with dimensionality 4 × Nvecs. A factorization of
the correlation function in terms of the eigenvectors of the Laplacian allows efficient
computation of correlation functions with a large basis of interpolating operators at
both the source and the sink time slices. Furthermore, this factorization also enables
a momentum projection at both the source and the sink time slices, in contrast with
what is possible in conventional spectroscopy. In this work, we used Nvecs = 64. It can
be shown that the correlation functions constructed using the distilled fields have same
symmetry properties on the lattice as those constructed using the Laplacian methods.
Variational fitting technique
Once we construct the matrix of two point correlation functions, Cij(t), for each lattice
irrep, we extract the energies and overlaps of the physical states over the interpolating
operators by solving a generalized eigenvalue problem of the form,
Cij(t)vnj = λn(t, t0)Cij(t0)vnj , (5)
where an appropriate reference time-slice t0 is selected following the description as in Ref.
[31]. The energies of the physical states are extracted by fitting the t − t0 dependence
of the eigenvalues, λn, called the principal correlators and the overlaps being determined
from the eigenvectors, vnj . The principal correlators are fit using a double exponential
form
λα(t) = e−mα(t−t0)(1 +O(e−|δm|(t−t0))), (6)
where mα is the mass of a state labeled by α and δm is the energy splitting from the
nearest excited state to α.
One main goal, which is a highly non-trivial task, in these lattice calculations is to
ensure that any state identified can be assigned continuum quantum numbers in a reliable
way. To identify the spin of a state we followed the same method detailed in Ref. [32]
and used in the calculations of light baryons [34, 35], light mesons [36], charm mesons
Synopsis xviii
as well as heavy-light mesons [37]. The main ingredient is the overlap factor, Zni , as
defined in eq. (1). However, spin-52
and spin-72
states appear in multiple lattice irreps.
In these cases, the continuum operator is subduced to multiple lattice irreps. In the
continuum limit, for a given physical state, the overlap factors obtained from different
operators, which are subduced from the same continuum operators should become equal.
They should nearly be equal even on a finite lattice, if the rotational symmetry breaking
effects are small. Hence in order to confirm the reliability of the identification of a state
with a given spin > 3/2, one has to compare the magnitudes of overlap factors of those
operators which are subduced into different irreps. After identification of the spin of
states with matching overlap factors, it is also necessary to check whether the energy of
this state also matches over the irreps. Once this is made, one needs to do simultaneous
fitting of its masses over the irreps to get the final mass.
0.3 RESULTS : Excited state spectroscopy of charm
baryons
In this section, we present our results on spectra of charm baryons with spins up to 72
and for both parities. Various spin dependent energy splittings between the extracted
states are also considered and compared with the similar splittings at light, strange as
well as bottom quark masses. We present our results in terms of energy splittings relative
to a reference hadron rather than their absolute energies. This reduces the systematic
uncertainties associated with charm quark mass determination as well as the with the
discretization errors.
0.3.1 Triply charm baryons
In Figure 1(a), we present relativistic spectra of the triply charm baryons in terms of
the energy splittings from the mass of the pseudoscalar charmonium, ηc, which is made
of a charm and an anti-charm quarks. A factor of 32
is multiplied with ηc mass to
balance the different number of valence charm quarks in triply charm baryon and ηc.
Boxes with the thick borders correspond to those states with a greater overlap onto
operators proportional to the field strength tensor as discussed in the previous section,
which might consequently be hybrid states. The states inside the magenta ellipses have
relatively large overlap with the non-relativistic operators. The number of states inside
Synopsis xix
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-0.0
0.4
0.8
1.2
1.6
2.0
2.4m
- 3
/2 m
ηc
[G
eV
]
Ωccc
(a)
|E(1/2 +) - E(3/2 +
)|
|E(5/2 +) - E(3/2 +
)|
|E(7/2 +) - E(3/2 +
)|
|E(5/2 +) - E(3/2 +
)|
|E(3/2 -) - E(1/2 -
)|
-160.0
-120.0
-80.0
-40.0
0.0
40.0
80.0
120.0
160.0
200.0
240.0
∆m
ass
(MeV
)
Hg,1,S
⊗ D[2]
L=2,SG
1g,1,M⊗ D
[2]
L=2,MG
1g,1,M⊗ D
[1]
L=1,M
Ωccc
(NR + R)
Ωbbb
(NR)
Ωccc
(NR)
Ωsss
∆uuu
(b)
Figure 1: (a) Spin identified spectra of triply charm baryons with respect to 32mηc . The
boxes with thick borders corresponds to the states with strong overlap with hybrid op-erators. The states inside the magenta ellipses are those with relatively large overlapto non-relativistic operators. (b) Energy splittings between states with same L and Svalues, starting from light to heavy baryons. The details are described in the Section0.3.1.
these ellipses are also the number of states expected based on a quark model with only
non-relativistic quark spins. This agreement of the pattern of low lying states between
the lattice spectra obtained in this work and from non-relativistic quark model implies a
clear signature of SU(6)×O(3) symmetry in the low-lying spectra. It is to be noted that
though we used non-relativistic as well as relativistic operators, the low-lying spectra
resembles with the non-relativistic spectra. Such SU(6) × O(3) symmetric nature of
spectra was also observed for light baryons in Ref. [35]. However, it is not meaningful
to interpret the higher excited states in terms of SU(6) × O(3) symmetry [35] since we
did not include non-relativistic operators with three derivatives. For negative parity, it
is not possible even to identify a state with strong hybrid content because it is unclear
how the relative importance of all the relevant operators overlapping to that state will
change in the presence of non-relativistic operators having three or more derivatives.
Energy splittings
Spin-dependent splittings, such as the hyperfine and spin-orbit splittings between hadrons,
can provide important information about the interquark interaction which are necessary
to build a successful model. Here we also calculated those splittings of triply charm
Synopsis xx
baryons and compare those with such splittings of similar baryons at other quark masses.
Among the states that we extract in our calculations, several of them have the same L and
S values, but with different J values. For example, we construct flavor (F ) decuplet states
with D = 2, S = 32
and L = 2 with the combination 10FS⊗(S)S⊗(D)S, where S in the
subscript stands for symmetric combinations. In this construction, we get four operators
with JP = 12
+, 3
2
+, 5
2
+and 7
2
+. In the heavy quark limit, the spin-orbit interaction van-
ishes since the interaction term is inversely proportional to the square of the quark mass.
States corresponding to quantum numbers (|L, S, JP 〉 ≡ |2, 32, 1
2
+〉, |2, 32, 3
2
+〉, |2, 32, 5
2
+〉and |2, 3
2, 7
2
+〉) will thus be degenerate in the heavy quark limit. Similarly, two states
with quantum numbers JP = 12
−and 3
2
−with L = 1 and S = 1
2will also be degenerate
in the heavy quark limit. In Figure 1(b), we plot absolute values of energy differences
between energy levels which originate from the spin-orbit interaction of the following
(L , S) pairs : (2,32
–in the left column), (2,12
–in the middle column) and (1,12
–in the
right column). We plot these spin-orbit energy splittings at various quark masses cor-
responding to following triply flavored baryons : ∆uuu, Ωsss, Ωccc and Ωbbb. We identify
the states with these (L, S) pairs by finding the operators which incorporate these pairs
and have major overlaps to these states. These energy differences are obtained from the
ratio of jackknifed correlators, which in general, reduce error bars by canceling correlation
between these correlators. For bottom baryons, we use data from Ref. [47] and for light
and strange baryons, results from Ref. [35] are used. One can observe that the degen-
eracy between these states is more or less satisfied both for bottom and charm quarks,
justifying a heavy quark status for charm and bottom quarks. However, data with higher
statistics are necessary to identify the breaking of this degeneracy at charm quark mass.
We also made similar comparisons for some other energy splittings using our results
and similar results at other quark masses. Some of these, such as the hyperfine splitting,
vanish in the heavy quark limit while others become constant. However, as expected,
most splittings tend to be higher at lighter quark masses where relativistic effects are
prominent. As an example, we consider following splittings: m∆uuu − 32mωuu ,mΩsss −
32mφss ,mΩccc − 3
2mJ/ψcc and mΩbbb − 3
2mΥbb
. For ∆++(uuu) and Ωsss baryons, results are
from Ref. [35], while for Ωbbb, we use results from Ref. [47]. These splittings mimic the
binding energies of triply flavored states and thus it is interesting to compare these as
a function of quark masses. One can expect that, in the heavy quark limit, the quark
mass dependence of these splittings can be expressed is a form a + b/mPS. We observe
such quark mass dependence starting from strange to bottom quarks. This validates the
Synopsis xxi
heavy quark expansion for these splittings, particularly for charm and bottom quarks.2
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
m -
mη
c
[G
eV
]
Ωcc
(a)
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
m -
mη
c
[G
eV
]
Ξcc
(b)
Figure 2: Spin identified spectra of doubly charm baryons, a) Ωcc and b) Ξcc, with respectto mηc . States with strong overlap with operators that are proportional to field strengthtensor are painted with thick borders. While the low lying bands showing the pattern ofstates as expected from an SU(6) ⊗ O(3) potential model are highlighted by enclosingthem within magenta closed curves.
0.3.2 Doubly charm baryons
The study of doubly charm baryons is important as these systems provide a unique
opportunity to get insight into the nature of strong force in the presence of slow relative
motion of the heavy quarks along with the relativistic motion of a light quark. The
excited spectra and the splittings between different energy levels will help to understand
how the collective degrees of freedom give rise excitations of these systems. A comparison
of their excitations with the corresponding spectra of singly and triply charm baryons,
where the number of charm quark is one less and more respectively, will be helpful to get
information about quark-quark interactions. Doubly charm baryons are characterized
by two widely separated scales: the low momentum scale, of order ΛQCD, of the light
quark and the relatively heavy charm quark mass. A doubly heavy baryon can be treated
as a bound state of a heavy antiquark and a light quark in the limit when the typical
momentum transfer between the two heavy quarks is larger than ΛQCD [48]. In this
limit of quark-diquark symmetry, QQq ↔ Qq, one can get definite prediction of spin
dependent energy splittings [49]. It is thus interesting to study those spin splittings to
check whether charm quark is heavy enough to respect this quark-diquark symmetry.
2The details have been skipped to save space.
Synopsis xxii
All previous lattice calculations involve only the ground state spectrum of Ξcc, Ξ∗cc,
Ωcc and Ω∗cc. However, it is expected that much more information about the interactions
between two charm quarks and between charm and light quarks can be obtained by
computing the excited state spectra of these baryons, including in particular the spin-
dependent energy splittings, as well as by studying similar spectra for other spin-parity
channels. Towards this goal, in this work, we make the first attempt to compute the
excited state spectra of doubly charm baryons by using dynamical lattice QCD. The
ground states for each spin-parity channel as well as their excited states up to spin 7/2
are computed and a few spin-dependent energy splittings are studied.
In Figure 2 we present the relativistic spectra of the doubly charm baryons, Ωcc(ccs)
and Ξcc(ccu), in terms of the energy splittings from the mass of ηc meson, which has
the same valence charm quark content. Even here, the number and the pattern of states
in the lowest negative parity band and the first excitation band in the positive parity
(magenta boxes in Figure 2) can be seen to be well in agreement with the expectations
based on SU(6)⊗O(3) symmetry, which assumes non-relativistic nature for quarks. As
before, the states with significant overlap with the operators proportional to the field
strength tensor are shown with thick borders, and are identified to be strongly hybrid in
nature.
Similar to the case of triply charm baryons, we estimate spin-dependent energy split-
tings, including hyperfine splittings, for doubly charm baryons, and then study the quark
mass dependence of some of those energy splittings. For some cases, we observe that the
quark mass dependence can also be fitted with a heavy quark inspired form a+ b/mPS.
From such quark mass dependencies we are able to make the following predictions :
mBc∗ −mBc = 80± 8 MeV, and mΩccb = 8050± 10 MeV.
0.3.3 Singly charm baryons
After triply and doubly charm baryons, we studied singly charm baryons with the same
details. However, for singly charm baryons the number of channels are many more,
namely, Λc(cud), Σc(cuu), Ξc(cus), and Ωc(css). In the presence of one or two light
quarks, the dynamics of these particles become much more complicated and these par-
ticles provide a unique opportunity to study the interaction between one or more light
quarks in the presence of a heavy quark. The number of possible interpolating operators
are also more as was shown in Table 3.
We show our results for the spectra of singly charm baryons in Figure 3 (name of each
Synopsis xxiii
particle channel are shown inside). As before, we plot spectra in terms of mass splittings
instead of absolute masses. In our calculation, the pion mass is mπ ∼ 391 MeV. Due
to this unphysical light quark mass, we plot the Λc(cud) and Σc(cuu) spectra in terms
of their splitting from the vector meson mass mρ(ud) calculated on these lattices. For
Ξc(cus), and Ωc(css), we show the spectra in terms of their splittings from the mass of
vector mesons K∗(us) and φ(ss), respectively. This set of splittings for singly charm
baryons is considered, since the leading order all the singly charm baryon spectrum have
the same discretization errors and the same effect of the light quark dynamics, making a
comparative study between the singly charm baryon spectra possible. It is to be noted
that with physical quark masses one may also opt to subtract D and Ds meson masses
instead, to reduce discretization error due to heavier charm quark.
As in the previous cases, we observed that the pattern of the extracted low-lying
states are remarkably similar to the expectation from a model with broken SU(6)×O(3)
symmetry. As previously, the low lying bands that have relatively large overlap with
the non-relativistic operators are shown with magenta boxes. Similarly the states with
significant overlap with the operators proportional to the field strength tensor are painted
with thick borders, and are identified as strongly hybrid in nature. Known physical states
are shown by violet stars (question marks are shown where quantum numbers are not
known). The mismatch between the experimental value and our calculated energies is
believed to be due to the unphysical quark masses and the finite volume effects.
The Σ++c and Ωc baryons have the same flavor structure : pqq. The allowed flavor
symmetries are similar to that of the doubly charm baryons, where one can have a
symmetric flavor structure and a MS/MA flavor structure. Thus the spectrum of these
two baryons should resemble the spectrum obtained for the doubly charm baryons. This
is quite evident from the two plots on the right in Figure 3.
The Λc baryons are those with quark content udc and can have an antisymmetric
or MS/MA flavor structure, which corresponds to Λc-singlet and Λc-octet respectively.
The difference between the Λc-octet and the Σ+c is that the latter is a part of the isospin
triplet, while the former forms an isospin singlet. The details of the flavor structure is
shown in Table 3.2. Since there is no totally antisymmetric spin combination that can
be constructed from a purely non-relativistic formalism, one requires angular momen-
tum structure through non-local behavior to construct a flavor-singlet Λc interpolating
operator. Figure 3(a) shows the spectrum of Λc, where one can see only one state in the
low lying band. This state has significant overlap with the flavors MS/MA with non-
relativistic spin 1/2. As observed for the other channels, the number and the pattern of
Synopsis xxiv
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
m -
mρ [
GeV
]
⊕
⊕
⊗
⊗
⊕⊗ ? ⊗ ?⊕
Λc
(a)
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
m -
mρ [
GeV
]
⊗⊕⊕⊗
⊗⊕ ?
Σc
(b)
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
m -
mK
* [
GeV
]
⊗⊕⊕
⊕⊗
⊗
⊕⊗ ? ⊗⊕ ?
Ξc
(c)
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
m -
mφ [
GeV
]
⊗⊕
⊕⊗
Ωc
(d)
Figure 3: Results on the spin identified spectra of (a) Λc, (b) Σc, (c) Ξc and (d) Ωc
baryons for both parities w.r.t. vector mesons, ρ (upper two) and K∗, φ (lower two).The keys are same as in Figure 1(a). Known physical states are shown by violet star(question marks are shown where quantum numbers are not known).
the first negative parity band and the first excited positive parity band agrees with the
non-relativistic predictions. It is to be noted that the ordering of the first two states,
particularly the lowest lying first excited positive parity state, is observed to have spin
5/2 which is similar to the pattern supported by experiments.
The Ξc baryons, with quark content usc, can have S, A or MS/MA flavor structure,
which correspond to Ξc-decuplet, Ξc-singlet and Ξc-octet respectively. Within Ξc-octet
flavor structure, one has two possibilities. If we assume an us-spin symmetry, similar to
the isospin that describes the symmetry between u and d quarks, there are two possible
MS/MA combinations : the Ξc-octet and the Ξ′c-octet. The latter is a part of the us-spin
triplet, of which Ωc and Σ++c are included, while the former is a us-spin singlet. As in
Synopsis xxv
the case of Λc, due to the absence of a totally antisymmetric spin combination that can
be constructed from a purely non-relativistic formalism, one requires angular momentum
structure through non-local behavior to construct a flavor-singlet Ξc interpolating oper-
ator. Hence the low lying spectrum for Ξc baryon will have two spin 1/2 states from the
Ξc-octet and Ξ′c-octet operators and a spin 3/2 state with the Ξc-decuplet flavor struc-
ture. This can clearly be seen in Figure 3(c). The full set of operators for Ξc baryon
forms a very large set that fails to converge and produce a reliable spectrum from the
variational fitting. In this analysis, hence we use purely non-relativistic operators for
fitting, and the Ξc spectrum shown in Figure 3(c) is non-relativistic.
0.4 Mixed action approach : Charm and strange
hadron spectroscopy
The main aim of this study is to explore heavy hadron spectra using the chiral overlap
fermions on the background of a set of improved gauge configurations generated with
highly improved staggered quarks. By using such an approach, we get an advantage to
use a chiral action as well as a large set of highly improved configurations with multiple
lattice spacings. Below I mention results obtained from my contribution in this project.
As mentioned in the introduction, this is a calculation complementary to the previous
study of excited charm baryon spectra, particularly to improve the control over the
discretization errors.
We use two sets of dynamical 2+1+1 flavors HISQ lattice ensemble generated by the
MILC collaboration : a set of 323 × 96 lattices with a =0.0877(10) fm and another set
of 483 × 144 lattices a =0.0582(5) fm. For valence quarks we use overlap action [50].
The strange and charm masses are set at their physical values while ml/ms = 1/5 for
both lattices. The details of these configurations are summarized in Ref. [41]. The above
lattice spacings are determined using the Ω baryon mass measured on these ensembles.
The results reported here are obtained from 110 configurations on the coarser lattice, and
65 configurations on the finer lattice. Using both point and wall sources, we calculate
various point-point, wall-point as well as wall-wall quark propagators to study the low
lying hadron spectra.
Our extracted pseudoscalar meson masses are within the range 400 − 5130 MeV
and 230 − 4000 MeV for the coarser and finer lattices, respectively. In following two
subsections, we will discuss our estimates for mesons and baryons mostly in terms of
Synopsis xxvi
(a)
Sec. 3[39] Sec. 4[27] Durr, et al.[28] PACS-CS[29]0.06
0.09
0.12
0.15
0.18
0.21
mass
Ωccc -
3/2
mass
J/ψ (
GeV
) cs=1.35
cs=2.0
a=0.0582fm
a=0.09fma=0.0888fm
a=0.07fm
(b)
Figure 4: (a) Meson mass splitting for charmonia and charm-strange mesons at twolattice spacings. Experimental values are shown in the left side. (b) Mass splitting of
the ground state of JP = 32
+Ωccc from 3/2mJ/ψ. Results of this mass splitting from this
work (blue) are compared with those obtained from the excited charm baryon studies inthe previous sections (red), Ref. [28] (violet) and Ref. [29] (green).
energy splittings as those have less systematics compared to extracted energies.
0.4.1 Energy splittings in charmonia and charm-strange mesons
In Figure 4(a) we present the energy splittings between axial, scalar and tensor charmonia
from pseudoscalar charmonia. In addition to this, we also calculate charm-strange mesons
with various quantum numbers. The energy splittings between these mesons are as
shown in Figure 4(b). We find that tuning of charm mass using kinetic mass brings
these splittings closer to their experimental values than when performed with the pole
mass [27]. Our results for the hyperfine splitting in 1S charmonia are 125(6) MeV and
119(3) MeV corresponding to coarser and finer lattices respectively. It is to be noted
that the this hyperfine splitting is one of the most well studied physical quantities in
lattice charmonium calculations over the year, and until very recently [15, 16], lattice
results were found to be smaller than the experimental value (∼ 116 MeV). Our results
are closer to the experimental value.
Charm baryons
We also study the ground state spectra of charm baryons with one or more charm quark
content, e.g., with quark content css, ccs, and ccc on these lattices. The mass splittings
Ωccc− 32J/Ψ are shown in Figure 4(b). The blue rectangles represent the results from this
Synopsis xxvii
work for the two different lattice spacings, while the rectangles in red are results from
the work discussed in the previous section. The two red points are with the tree-level
clover coefficient cs = 1.35 and at a boosted value of cs = 2, which gives correct char-
monium hyperfine splitting, J/ψ − ηc. The red and blue squares showed the consistency
between our two independent calculations. These results are also consistent with other
independent lattice results [28, 29]. We also calculate various spin splittings for doubly
charm baryons and find similar consistency between our two results and results from
other lattice calculations.
0.5 Baryon screening masses at finite temperature
0.5
0.6
0.7
0.8
0.9
1
1.1
PS SC V AV N+ N−
µh/µ
hF
FT
clover 2011 [42]clover 2006 [55]
overlap 2008 [56]staggered 1988 [58]staggered 2002 [57]
Figure 5: The ratio of screening massesmeasured at T = 1.5Tc in quenched QCDwith those in FFT. Although all the re-sults are not taken for exactly the samequark mass, the effects of the quark massare very small (less than 1% in this study).
mπTc
µN+
mN+
µN−−µN+
mN−−mN+
2.20 1.05± 0.05 0.75± 0.10
1.99 1.07± 0.04 0.68± 0.12
Table 4: Thermal shifts in nu-cleon masses and mass splittings atT = 0.95Tc; µN+/N− denotes the in-verse screening length in the posi-tive/negative parity nucleonic channel,and mN+/N− the respective mass at T =0.
The long term experimental programs on finite temperature QCD are beginning to
probe the finer details of the non-perturbative predictions of QCD [51], at various collider
experiments, including Relativistic Heavy Ion Collider (RHIC) at BNL and Large Hadron
Collider (LHC) at CERN. This is possible because fireballs produced in these collisions
come close to thermal equilibrium [52] and the data from such experiments need to be
confronted with first principles calculations such as Lattice QCD techniques that can
make useful theoretical predictions on the expectation value of observables.
Synopsis xxviii
Using lattice QCD, the phases and properties of the medium can be studied by ana-
lyzing the static correlation lengths (which is inverse of the screening mass, µ) [53]. These
correlations are measured by introducing static probes into the equilibrium plasma and
measuring the response of the medium. The response depends on the quantum numbers
carried by the probes; so one can classify static correlators as glueball-like, meson-like
and baryon-like probes with the usual quantum numbers of these quantities corrected
for the fact that the static spatial symmetries are different from the Poincare group.
In this work, we study the meson-like and baryon-like static correlation lengths above
(1.5 Tc) and immediately below (0.95 Tc) the deconfinement transition temperature. It
is expected that these studies provide important inputs to the study of baryon-number
fluctuations, which is of much interest in the experimental search for the critical point of
QCD. We reboot the study of these correlators in the high temperature phase.
At 1.5 Tc, we find that the scalar/psuedoscalar (S/PS) and the (vector/axialvector)
V/AV correlators become pairwise degenerate. We find strong signals of approximate
chiral symmetry restoration from the near degeneracy of the screening masses of the
mesonic parity partners. Figure 5 shows the ratio of screening masses of various hadrons
in an interacting theory to the respective screening masses in free field theory. In addition
we found, as before [54, 55, 56], that the screening masses are not far from those obtained
in Free Field Theory (FFT), which is a model of non-interacting quarks. Along with our
estimates for the ratio of screening masses in interacting theory to the free theory, Figure
5 also contains a survey of the literature. This work uses the smallest lattice spacing used
in a quenched study to date, and agrees with previous work using clover quarks [55]. It
also agrees with results from studies using overlap quarks at smaller lattice spacing [56].
However, it seems that studies with naıve staggered quarks always indicate stronger
deviations from FFT than any of these above studies [57, 58]. Immediately below Tc,
we find that the mass of the mesons is hardly affected by the temperature below the
deconfinement transition temperature.
As in the mesonic correlations, the nucleonic screening lengths at T = 1.5 Tc, show
strong signals of approximate chiral symmetry restoration through the near degeneracy
of their screening masses. It is also evident from Figure 5 that above Tc the baryonic
screening lengths also support the evidence for weakly interacting quarks. However, be-
low Tc some interesting observations can be seen from Table 4. We find that µN+= mN+
within 95% confidence limits, indicating no significant thermal effects in the nucleon
mass. However, the splitting between the N+ and its parity partner, N−, changes at
finite temperature, and is a precursor to the restoration of chiral symmetry before the
Synopsis xxix
QCD phase transition. These observations not only constrain models of quantum hadro-
dynamics [59], but could also have implications for the analysis of heavy-ion collision
data.
0.6 Summary and conclusions
In this synopsis, we summarized our study of baryons by using non-perturbative methods
of lattice QCD. The significant part of the work dealt with spectroscopy of charm baryons.
In this, we performed two complementary studies using two different non-perturbative
techniques with distinct merits, and comparative study was performed between the results
from both these studies, which gave us confidence in our results. A second part of this
synopsis involved a study of baryon screening masses at finite temperature that could
provide important information on the properties of the QCD medium across the transition
temperature.
The detailed study of charmonia, both theoretically and experimentally, has provided
crucial insights in our understanding of strong interaction. In contrast to that, charm
baryons have not been studied comprehensively, though that can also provide similar
information. In this work, we make the first attempt to study charm baryons comprehen-
sively. An anisotropic action with 2+1 flavor dynamical clover and with temporal lattice
spacing of about (5.67 GeV)−1 was used. A large set of carefully designed baryonic inter-
polating operators with up to two derivatives were constructed and the newly developed
distillation technique was employed to generate the correlation functions. A well-suited
variational technique was utilized to extract the energy eigenstates with reliable spin-
parity quantum numbers. The main results are shown in Figure 1, 2 and 3. Among the
results most significant ones are the following predictions: mΩccc −mJ/ψ = 160± 6 MeV,
mBc∗ − mBc = 80 ± 8 MeV, and mΩccb = 8050 ± 10 MeV. In the extracted spectra we
found bands of states with alternating parities and increasing energies. Beside identify-
ing the spin of a state we could also decode the structure of operators leading to that
state : whether constructed by relativistic, non-relativistic, hybrid, non-hybrid types or
a mixture of them all. However, for negative parity states and highly excited positive
parity states, this identification is not possible since we do not include operators with
more than two derivatives which will contribute to these states and so will change the
relative contribution from various operators leading to such states. Similar to light and
strange baryon spectra [35], we also found the number of extracted states of each spin
in the three lowest-energy bands and the number of quantum numbers expected based
Synopsis xxx
on weakly broken SU(6) × O(3) symmetry agree perfectly, i.e., all the low lying charm
baryon spectra remarkably resemble the expectations of quantum numbers from quark
model [6, 7, 8] though we used relativistic as well as non-relativistic hybrid operators
which are not present in such models.
In addition to the pattern of the spectra, we also studied various energy splittings that
could give insights into spin-dependent interactions that exists between multiple charm
quarks as well as charm quarks and the light or the strange quarks. In the heavy quark
limit, the energy splitting due to spin-orbit coupling should vanish as it is proportional
to the heavy quark masses. To check this, we identify a few states with the same L
and S values from overlap factors of various operators and found that the spin-orbit
energy degeneracy between these states are more or less satisfied both for bottom and
charm quarks, justifying a heavy quark status for the charm quarks. More precise data
are necessary to check the breaking of this degeneracy at the charm quark mass. We
also evaluated some other energy splittings that mimics the binding energy of various
baryon states, such as : mΩbbb − 32mΥbb
,mΩccc − 32mJ/ψcc and mΩsss − 3
2mφss . Significant
quark mass dependence was observed for such splittings that can be modeled with a form
a + b/mPS, which assumes they will tend to a constant in the heavy quark limit. It is
interesting to note that the heavy quark expansion gives a good fit with data at bottom,
charm as well as strange masses.
It is to be noted that we did not use any multi-hadron operators in this calculation.
Inclusion of those operators, particularly those involving light quarks, may affect some of
the above conclusions, though to a lesser extent than their influence in the light hadron
spectra. Also this calculation was carried out with one lattice spacing, one volume
and at mπ = 391 MeV. In order to get precise estimate, which can be compared with
experiments, this study needs to be repeated with multiple volumes and lattice spacings
at lighter pion masses.
Complementary to the above work, we calculated charm and strange hadron spectra
with the overlap fermions in the background of a set of 2+1+1 flavors HISQ gauge
configurations. By adopting such a mixed action approach, we took advantage of the
chiral symmetry and low quark mass limit of overlap fermions, and the advantage of
having a large set of configurations with small discretization errors as well as small
taste breaking effects. We were also able to simulate both light, strange as well as
heavy fermions on the same lattice formalism with chiral fermions having no O(a) errors.
Results obtained, in particular the splittings between various hadrons, including the
hyperfine splitting in 1S charmonia, are encouraging and consistent with experiments.
Synopsis xxxi
This suggests that the overlap valence quarks on 2+1+1 flavor HISQ configurations is a
promising approach to do lattice QCD simulation with light, strange and charm quark
together in the same lattice formulation. We found the tuning of the charm quark mass
with the kinetic mass, rather than the pole mass, improves the discretization errors. We
also have the estimates for various charm and charm-strange hadrons. The splitting
(mΩccc − 32mJ/Ψ), between J/Ψ and the unknown triply charm baryon Ωccc was found to
be 145(10) MeV and 144(10) MeV, on our coarser and finer lattices respectively.
This is a complementary study of the first project to access the effect of discretization
errors on extracted masses. In both the works, we calculated the Ωccc mass, for which the
discretization error is maximum since all the three quarks are heavy. We observed that
results from both the calculations are consistent with each other and are also consistent
with a few other independent lattice calculations. This shows that the discretization
errors entering in our estimates are in good control.
It is to be noted that we have not performed continuum and chiral extrapolations
which will be performed later by ILGTI after adding another lattice spacing. The chiral
extrapolation has to be done carefully by using mixed action partially quenched chiral
perturbation theory [60] with proper mixed action discretization term ∆mix [61].
In the second part of the work, we calculated thermal correlations in quenched QCD
in channels with various hadronic quantum numbers, with a aim to access the behavior
of baryons at finite temperature. The correlation functions are consistent with the usual
picture of the QCD phase diagram : below Tc long-distance correlations are mediated
by hadrons and above Tc these are mediated by weakly interacting quarks. Above Tc,
at the temperature T = 1.5Tc, we found strong signals of approximate chiral symmetry
restoration in the near degeneracy of screening masses of hadronic parity partners. In
addition, we found, as before [54, 55, 56], that the correlation functions and screening
masses are not far from those obtained in FFT, which is a model of non-interacting quarks.
The estimates from this work is in consistence with previous results using clover quarks
[55] and overlap quarks [56], while staggered quarks in the quenched approximation
always indicated stronger deviations from FFT [57, 58].
Mesonic channels and the nucleon ground state show no thermal effects below Tc.
However, the opposite parity excitation of the nucleon, S11, is seen to move closer to the
ground state signaling a precursor to chiral symmetry restoration. No such effects have
been seen earlier either in the glue sector or with quarks, nor do we see any effect in the
V/AV channels here. This is a new observation which can constrain models of quantum
hadrodynamics, as well as have implications for the analysis of heavy-ion collision data.
Synopsis xxxii
This encourages for future study of baryonic correlators in a comprehensive way.
Bibliography
[1] P. W. Higgs, Phys. Lett. 12, 132 (1964);
S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716, 30 (2012)
[arXiv:1207.7235 [hep-ex]];
G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716, 1 (2012) [arXiv:1207.7214
[hep-ex]].
[2] RPP : J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012).
[3] M. Mattson et al. [SELEX Collaboration], Phys. Rev. Lett. 89, 112001 (2002) [hep-
ex/0208014].
[4] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 74, 011103 (2006) [hep-
ex/0605075].
[5] Y. Kato et al. [Belle Collaboration], arXiv:1312.1026 [hep-ex].
[6] S. Capstick and W. Roberts, Prog. Part. Nucl. Phys. 45, S241 (2000) and the refer-
ences therein.
[7] E. Klempt and J. -M. Richard, Rev. Mod. Phys. 82, 1095 (2010) and the references
therein.
[8] V. Crede and W. Roberts, Rept. Prog. Phys. 76, 076301 (2013) and the references
therein.
[9] J. D. Bjorken, FERMILAB-CONF-85/69, Is the ccc a New Deal for Baryon Spec-
troscopy?, Int. Conf. on Hadron Spectroscopy, College Park, MD, Apr. 1985;
[10] S. Aoki, et al. [CP-PACS Collaboration], Phys. Rev. Lett. 84, 238 (2000); Phys.
Rev. D 79, 034503 (2009).
[11] C. W. Bernard, et al., Phys. Rev. D 64, 054506 (2001) [hep-lat/0104002].
xxxiii
Synopsis xxxiv
[12] C. T. H. Davies et al. [HPQCD and UKQCD and MILC and Fermilab Lattice Col-
laborations], Phys. Rev. Lett. 92, 022001 (2004) [hep-lat/0304004].
[13] C. Aubin, et al., Phys. Rev. D 70, 094505 (2004) [hep-lat/0402030].
[14] S. Durr, et al., Science 322, 1224 (2008) [arXiv:0906.3599 [hep-lat]].
[15] T. Burch, et al., Phys. Rev. D 81, 034508 (2010).
[16] G. C. Donald, et al., Phys. Rev. D 86, 094501 (2012).
[17] Y. Namekawa et al. [PACS-CS Collaboration], Phys. Rev. D 84, 074505 (2011).
[18] C. McNeile, et al., Phys. Rev. D 86, 074503 (2012) [arXiv:1207.0994 [hep-lat]].
[19] D. Mohler and R. M. Woloshyn, Phys. Rev. D 84, 054505 (2011);
D. Mohler, S. Prelovsek and R. M. Woloshyn, Phys. Rev. D 87, 034501 (2013).
[20] R. J. Dowdall, et al., [HPQCD Collaboration], Phys. Rev. D 85, 054509 (2012)
[arXiv:1110.6887 [hep-lat]]; Phys. Rev. D 86, 094510 (2012) [arXiv:1207.5149 [hep-
lat]].
[21] E. B. Gregory, et al., Phys. Rev. D 83, 014506 (2011) [arXiv:1010.3848 [hep-lat]].
[22] N. Mathur, R. Lewis and R. M. Woloshyn, Phys. Rev. D 66, 014502 (2002);
R. Lewis, N. Mathur and R. M. Woloshyn, Phys. Rev. D 64, 094509 (2001).
[23] J. M. Flynn, et al., [UKQCD Collaboration], JHEP 0307, 066 (2003) [hep-
lat/0307025].
[24] L. Liu, et al., Phys. Rev. D 81, 094505 (2010) [arXiv:0909.3294 [hep-lat]].
[25] G. Bali, et al., J. Phys. Conf. Ser. 426, 012017 (2013) [arXiv:1212.0565 [hep-lat]].
[26] R. A. Briceno, H. -W. Lin and D. R. Bolton, Phys. Rev. D 86, 094504 (2012)
[arXiv:1207.3536 [hep-lat]].
[27] S. Basak, et al., [ILGTI] PoS LATTICE 2012, 141 (2012) [arXiv:1211.6277 [hep-lat]];
PoS LATTICE 2013, 243 (2013) [arXiv:1312.3050 [hep-lat]].
[28] S. Durr, G. Koutsou and T. Lippert, Phys. Rev. D 86, 114514 (2012)
[arXiv:1208.6270 [hep-lat]].
Synopsis xxxv
[29] Y. Namekawa et al. [PACS-CS Collaboration], Phys. Rev. D 87, 094512 (2013).
[30] M. Peardon, et al., [Hadron Spectrum Collaboration], Phys. Rev. D 80, 054506
(2009).
[31] C. Michael, Nucl. Phys. B 259, 58 (1985);
M. Luscher and U. Wolff, Nucl. Phys. B 339, 222 (1990).
[32] J. J. Dudek, R. G. Edwards, N. Mathur and D. G. Richards, Phys. Rev. D 77,
034501 (2008).
[33] S. Basak et al. [Lattice Hadron Physics (LHPC) Collaboration], Phys. Rev. D 72,
074501 (2005).
[34] R. G. Edwards, J. J. Dudek, D. G. Richards and S. J. Wallace, Phys. Rev. D 84,
074508 (2011).
[35] R. G. Edwards, N. Mathur, D. G. Richards and S. J. Wallace, Phys. Rev. D 87,
054506 (2013).
[36] J. J. Dudek, et al., Phys. Rev. Lett. 103, 262001 (2009); Phys. Rev. D 82, 034508
(2010).
[37] L. Liu, et al., [HSC], JHEP 1207, 126 (2012);
G. Moir, et al., JHEP 1305, 021 (2013).
[38] J. J. Dudek and R. G. Edwards, Phys. Rev. D 85, 054016 (2012) [arXiv:1201.2349
[hep-ph]].
[39] M. Padmanath, R. G. Edwards, N. Mathur and M. Peardon, arXiv:1307.7022 [hep-
lat];
arXiv:1311.4354 [hep-lat]; arXiv:1311.4806 [hep-lat].
[40] E. Follana et al. [HPQCD and UKQCD Collaborations], Phys. Rev. D 75, 054502
(2007).
[41] A. Bazavov et al. [MILC Collaboration], Phys. Rev. D 87, no. 5, 054505 (2013).
[42] S. Datta, S. Gupta, M. Padmanath, J. Maiti and N. Mathur, JHEP 1302, 145
(2013).
Synopsis xxxvi
[43] K. G. Wilson, Phys. Rev. D 10, 2445 (1974).
[44] G. Burgers, F. Karsch, A. Nakamura and I. O. Stamatescu, Nucl. Phys. B 304, 587
(1988); T. R. Klassen, Nucl. Phys. B 533, 557 (1998); P. Chen, Phys. Rev. D 64,
034509 (2001).
[45] R. G. Edwards, B. Joo and H. -W. Lin, Phys. Rev. D 78, 054501 (2008)
[arXiv:0803.3960 [hep-lat]].
[46] H. -W. Lin et al. [HSC], Phys. Rev. D 79, 034502 (2009) [arXiv:0810.3588 [hep-lat]].
[47] S. Meinel, Phys. Rev. D 85, 114510 (2012).
[48] M. J. Savage and R. P. Springer, Int. J. Mod. Phys. A 6, 1701 (1991);
M. J. Savage and M. B. Wise, Phys. Lett. B 248, 177 (1990).
[49] N. Brambilla, A. Vairo and T. Rosch, Phys. Rev. D 72, 034021 (2005) [hep-
ph/0506065];
S. Fleming and T. Mehen, Phys. Rev. D 73, 034502 (2006) [hep-ph/0509313].
[50] H. Neuberger, Phys. Lett. B 417, 141 (1998); ; ibid. B427 (1998) 353;
M. Luscher, Phys. Lett. B 428, 342 (1998).
[51] S. Gupta, et al., Science 332, 1525 (2011).
[52] J. Cleymans and K. Redlich, Phys. Rev. Lett. 81 (1998) 5248;
P. Braun-Munzinger and J. Wambach, Rev. Mod. Phys. 81 (2009) 1031.
[53] C. E. DeTar and J. B. Kogut, Phys. Rev. Lett. 59 (1987) 399, Phys. Rev. D 36
(1987) 2828.
[54] K. D. Born et al.(MTc Collaboration), Phys. Rev. Lett. 67 (1991) 302.
[55] S. Wissel et al., PoS LAT-2005, 164; S. Wissel, Ph. D dissertation (2006).
[56] R. V. Gavai, S. Gupta and R. Lacaze, Phys. Rev. D 78 (2008) 014502.
[57] R. V. Gavai, S. Gupta and P. Majumdar, Phys. Rev. D 65 (2002) 054506.
[58] A. Gocksch, P. Rossi and Urs M. Heller, Phys. Lett. B 205 (1988) 334.
[59] R. Rapp and J. Wambach, Adv. Nucl. Phys. 25 (2000) 1.
Synopsis xxxvii
[60] O. Bar, G. Rupak and N. Shoresh, Phys. Rev. D 67, 114505 (2003);
J. -W. Chen, D. O’Connell and A. Walker-Loud, Phys. Rev. D 75, 054501 (2007);
K. Orginos and A. Walker-Loud, Phys. Rev. D 77, 094505 (2008) and references
therein.
[61] M. Lujan, et al., Phys. Rev. D 86, 014501 (2012).
Publications
• S. Basak, S. Datta, A. T. Lytle, M. Padmanath, P. Majumdar and N. Mathur,
PoS LATTICE 2013, 243 (2013), [arXiv:1312.3050 [hep-lat]].
• M. Padmanath, R. G. Edwards, N. Mathur and M. Peardon, arXiv:1311.4806
[hep-lat], conference proceedings : CHARM 2013.
• M. Padmanath, R. G. Edwards, N. Mathur and M. Peardon, PoS LATTICE
2013, 247 (2013), arXiv:1311.4354 [hep-lat].
• M. Padmanath, R. G. Edwards, N. Mathur and M. Peardon, arXiv:1307.7022
[hep-lat], submitted to Phys. Rev. D.
• S. Datta, S. Gupta, M. Padmanath, J. Maiti and N. Mathur, JHEP 1302, 145
(2013) [arXiv:1212.2927 [hep-lat]].
• S. Basak, S. Datta, M. Padmanath, P. Majumdar and N. Mathur, PoS LATTICE
2012, 141 (2012) [arXiv:1211.6277 [hep-lat]].
xxxviii
Outline of the thesis
In this thesis, I present my work on the non-perturbative studies of various aspects of
baryon physics using lattice QCD. An outline of the works presented in the thesis is as
follows :
• Spectroscopy of charm hadrons
– Excited state charm baryon spectroscopy using clover fermions.
– Charm hadron spectroscopy using overlap fermions.
• Nucleons at finite temperature.
The organization of this thesis is as described below.
In Chapter 1, I introduce the aforementioned works, covering the phenomenology in
the respective aspects. I discuss the experimental and the theoretical advances in the
spectroscopic studies and the future prospects from both experimental and theoretical
point of view. After the phenomenological background and motivation, I discuss the
works on Excited state charm baryon spectroscopy and Charm hadron spectroscopy using
overlap fermions. The work on Nucleons at finite temperature is discussed briefly in the
last section.
Chapter 2 serves as very brief review of the basics of the first principle non-perturbative
technique, known as Lattice QCD, that we employ for our calculations. The review cov-
ers a short introduction to the QCD in the continuum followed by a section dedicated
to the lattice regularization of QCD. A third section discusses the basics of simulations
involved and an introduction to the conventional spectroscopy studies using lattice.
Chapter 3 is dedicated for the technical details related to the work on excited state
spectroscopy. I begin by discussing the procedure involved in conventional spectroscopy
studies using lattice QCD and the challenges that appear in the excited state spec-
troscopy. Individual sections are dedicated to explain the solutions, which include the
xxxix
Outline of the thesis xl
use of anisotropic lattice formalism, derivative based operator construction formalism
and variational fitting techniques, to these challenges.
In Chapter 4, our results for charm baryons from the excited state spectroscopy
studies are presented. The chapter is divided into three sections : 1) Triply charm
baryons, 2) Doubly charm baryons and 3) Singly charm baryons and spectra for them
are discussed in the respective sections. Along with I also discuss our calculations to
assess the discretization errors entering our estimates. Separate subsections are used for
various energy splittings, which carry the information about the binding energies of the
extracted states. We also discuss various systematics involved in this calculation that
have not been addressed.
In Chapter 5, I discuss the details and the results from our heavy hadron spectroscopy
calculations using overlap fermions. In this chapter, we discuss the detailed study of the
discretization errors in this non-perturbative approach, and make a comparative study
of various estimates that we measure from this work and the previous one, along with
estimates from other lattice and potential model calculations.
In Chapter 6, I discuss our study of hadron screening masses at finite temperature.
Main emphasis has been given for the nucleonic channels, where we observe interesting
behavior in the negative parity nucleonic channels indicating precursor effects for the
chiral symmetry restoration in the nucleonic channels immediately below Tc.
In Chapter 7, we give a brief summary of the findings made in the works described
in the thesis.
Chapter 1
Introduction
Understanding the building blocks of the universe and the fundamental laws of nature
that govern the interactions between them is the primary aim of theoretical physics, with
a hope that this understanding will help us explain all the complex natural phenomena
around us. This view is known as reductionism. Success of such a view in atomic physics
began with John Dalton’s (1803) atomic theory, followed by Dmitri Mendeleev’s proposal
of atomic periodic table and the developments that followed till the early 20th century.
Atomic physics further got a substantial boost in its understanding about the atomic
spectra and atomic substructure following Ernest Rutherford’s Gold foil experiment,
Niels Bohr’s proposal of his atom model and, in particular, the development of quantum
mechanics. The enormity of the success of quantum mechanics is such that it found
applications in a wide variety of areas beyond atomic physics, including a great deal of
modern technological inventions which revolutionized our life.
Particle physicists also believe in such a reductionist point of view in order to un-
derstand the proliferated collection of particles that were observed after the invention of
cyclotron and the development of various particle accelerators. The rich spectra of light
hadrons provided us with a framework for constructing a theory of strong interactions
starting from the eightfold way to treating quarks and gluons as the fundamental degrees
of freedom. The development including the unification of the electromagnetic and the
weak interactions [1, 2, 3], followed by the proposal of Higgs field [4], which is believed
to give mass to all the fundamental particles in the universe, gave the Standard Model
of particle physics its modern form.
Though the discoveries of the neutral intermediate vector boson, the top quark, the
τ -neutrino and the Higgs boson [5, 6] put the Standard Model of particle physics on solid
1
Introduction 2
footing, there remains significant limitations in our understanding. Quantum Chromo
Dynamics (QCD), a local non-Abelian gauge theory representing the SU(3) sector in
the Standard Model, is established to be the theory of strong interactions. The most
desirable feature of QCD, the asymptotic freedom [7, 8], allowed successful application
of perturbative techniques in explaining the observations in the high energy domain from
various particle collider experiments. However, a complete understanding of the non-
perturbative part of QCD in the low energy domain remains a challenge. The masses of
the hadrons made of light quarks, in particular baryons, are observed to be an order of
magnitude higher than the sum of the mass of the quarks that constitutes them. Baryons,
particularly neutrons and protons, account for ∼99% of the mass of the visible universe.
However, only ∼1% of the proton mass is explained by the Higgs mechanism. Hence
it is very important to understand the origin of the mass that constitutes the major
part of the visible universe. From various lattice studies, it is now understood that this
mass comes from the non-perturbative nature of the strong interactions in the low energy
domain. The calculation of the energy spectra of various hadrons is the main aim of this
work.
1.1 Baryons
Baryons are relatively simple systems in which quintessentially confining character of
multiple quarks is manifest and are sufficiently complex to shed light on physics hidden
from us in the mesons. Understanding the baryon spectra is thus expected to answer
a large series of questions on baryons and the interactions that exist within them [9].
A few questions for which the current day and the future baryon spectroscopy studies
are expected to provide answers are the following. Can all baryon resonances be ex-
plained from QCD? What are the leading interactions that govern the spectra of baryons
from the light to the heavy sector? Can we find signatures for the property of flavor
independence of the QCD interactions? How much the mean field theories, like quark
model calculations that are highly successful in explaining the low lying spectrum, will be
successful in explaining the full resonance spectrum? Why are the ‘missing resonances’
missing? Missing resonances are those that are predicted by symmetric quark models
but have not been observed. Even the diquark models, in which the diquark excitations
are suppressed, predicts a lot more resonances than observed. Do they really exist in
nature? What is the spectral pattern of the highly excited states? Are they organized
in the form of spin-parity doublets or chiral multiplets of mass degenerate states having
Introduction 3
identical spin and parity?
Though in its infancy, one also expects to understand a number of questions in the
heavy baryon sector. The triply heavy baryons are expected to shed light into the quark
dynamics in baryons, which is somewhat hidden by the chiral dynamics in the light
baryons. The doubly charm spectrum will explain if it is dominantly governed by a
charmonium-like heavy quark dynamics or a charm meson-like relativistic light quark
dynamics around a static color source formed by the heavy-heavy diquark. While the
spectra of singly charm baryons are expected to help us understand the hierarchy of the
light quark excitations. Particularly, with the recent excitement in heavy hadron physics
and fore-seeing the observations from large statistical samples that will be collected in
many future and ongoing experiments [10, 11, 12, 13, 14, 15], a detailed understanding
of the heavy baryon spectra is expected to provide insights into various aspects of the
strong force that cannot be probed with light baryons.
1.2 Heavy hadrons in QCD
A comprehensive understanding of the theory of QCD in the strong coupling regime
implies a rigorous determination and understanding of its bound states. If QCD is the
correct theory, then one should be able to reproduce the physical hadron spectra from a
high precision first principles non-perturbative calculation. Much like the role of atomic
spectroscopy in the development of quantum mechanics, hadron spectroscopy, both the-
oretically and experimentally, has played a crucial role in understanding the nature of
the fundamental strong force and its degrees of freedom. The dense spectra of hadrons
with light quarks provided us with a framework for constructing the theory of strong
interaction. The discovery of the heavier hadron, J/ψ meson, and the subsequent dis-
coveries of other charmonia put this framework on a solid footing. These discoveries
and the identification of these states as charm-anti-charm bound states took the con-
cept of quarks from a mere mathematical construct to reality. The discovery of J/ψ
meson triggered such a huge scientific interest that it was termed as the November revo-
lution. The charm quark mass being heavy, the approximation that the charm quark and
anti-quark are non-relativistic were found to be good for many predictions. Studies on
various energy splittings provided crucial information on heavy quark-anti-quark poten-
tial, hyperfine and spin-orbit interactions, etc. The ground state charmonia, J/Ψ and ηc,
gave the excellent analogs of hydrogen atom for strong interactions, which revealed the
quark layer of substructure for hadrons. Potential models, consisting of non-relativistic
Introduction 4
quark kinetic energy, a central confining potential, and spin-dependent interaction terms
were found to be very successful in explaining various observed states. Recently, a tower
of heavy hadron states, including the X’s, Y ’s and Z’s, has been discovered with un-
usual properties [16]. These discoveries have rejuvenated the heavy hadron spectroscopy
tremendously and offer a bright promise in detailed understanding of various interesting
aspects of the theory of strong interaction.
While the heavy quarkonia and other heavy flavored mesons have been studied quite
extensively, the heavy baryon physics has received substantially less attention, even
though they can provide similar insight in understanding the strong interaction. Ex-
perimentally nearly a score of singly charm baryons have been discovered, but only a
few of the low lying states have been assigned the spin-parity quantum numbers based
on experimental evidence [16]. A reliable determination of the quantum number of most
of the observed states has not been made. For doubly charm baryons, only experiments
at SELEX reported the discovery of five doubly charm baryon resonances and inter-
preted those as Ξ+ccd(3443), Ξ+
ccd(3520), Ξ++ccu (3460), Ξ++
ccu (3541) and Ξ++ccu (3780) [17, 18].
Later they confirmed the Ξ+ccd(3520) state in two different decay modes (Ξ+
cc → ΛcK−π+;
Ξ+cc → pD+K−) at a mass of 3518.7 ± 1.7 MeV with an average lifetime of less than 33
fs [19]. However, these states have not been observed either by BABAR [20] or Belle
[21, 22] in e+e− annihilation experiments. Along with the well-established triply fla-
vored ∆(uuu) and strange Ω(sss) baryons, QCD predicts similar states built from charm
quarks, the triply charm baryon, Ωccc. Such a state is yet to be observed. This could be
because of the large energy threshold required for their production via either resonant
or continuum production mechanisms, their very short lifetime and the very low recon-
struction efficiency for the highly excited heavy hadron resonances, which follow cascade
decays into multi-particle final states. The extremely low production rate hinders the
identification of the spin-parity quantum numbers, even though the partial wave analysis
is relatively simple in heavy hadrons in comparison with the light hadron spectra. Being
a baryon-baryon collider experiment, SELEX, unlike BaBar and Belle, may have had
huge cross section for the production of the charm baryons through continuum produc-
tion mechanism from the QCD background. This could be a reason for the observation
of the doubly charm baryon in SELEX experiments, while not in BaBar or Belle. One of
the main aims of the proposed PANDA experiment at the Facility for Anti-proton and
Ion Research (FAIR), Darmstadt, is to provide an ideal environment to study the heavy
hadron physics. It will investigate anti-proton annihilations in the momentum range from
1.5 GeV to 15 GeV. The ongoing proton-proton collider experiments at LHCb aim at wide
Introduction 5
range of physics programs covering many important aspects of heavy flavor physics. The
high intensity baryon beams and the gluon rich reaction product from the baryon-baryon
and the nucleus-nucleus collisions in both these experiments will allow high precision
spectroscopy of heavy hadrons.
1.2.1 Potential model calculations
On the theoretical side, one expects that potential models will be able to describe, charm
baryons [9, 23, 24], in particular the triply charm baryons, to a similar level of precision
as their success in charmonia. The triply charm baryons may provide a new window
for understanding the structure of baryons, as pointed out by Bjorken several years ago
[25]. Just as the quark-anti-quark interactions are examined in charmonia, these studies
can probe the interactions between multiple heavy quarks and heavy quarks with one
or more lighter quarks. Various spin dependent splittings between baryons can provide
information to constrain spin dependent potential terms, which are crucial in building
successful Non-Relativistic QCD (NRQCD), pNRQCD and similar models [26, 27]. The
low lying spectra of charm baryons have been studied theoretically using various ap-
proximate formulations [9, 23, 24] of QCD. These include potential model calculations
with non-relativistic and relativistic heavy quarks utilizing various symmetries in a heavy
quark system, calculations based on QCD sum rules and Regge phenomenology. These
kind of calculations were performed as early as in late 1970’s by Isgur and Karl [28, 29]
where they studied baryons in a quark model framework inspired by QCD. These were
followed by studies using a QCD motivated bag model [30]. Potential model calculations
in non-relativistic quarks were performed in a number of works using simple variational
methods [31] or by employing Faddeev formalism to solve the three body problem of
baryons [32, 33, 34]. Taking into account the presence of the valence heavy flavor, calcu-
lations were performed based on heavy quark effective theory formulations [35], and with
heavy quark spin symmetry constraints [36, 37, 38] to solve the non-relativistic three
body problem. In Ref. [39] masses and mass relations between heavy baryons with one
heavy quark using 1/mQ and 1/Nc expansions were studied. Non-relativistic potential
model calculations were also performed treating baryons as quark-diquark systems [40],
where the excitations in the diquark are frozen or occur at significant higher energies.
Calculations were also performed using NRQCD and its improvisations called pNRQCD
formalisms, which were found to be successful in classifying the low lying charmonia and
bottomonia [41, 42, 43, 44]. Owing to the fact that the NRQCD Lagrangian can be
Introduction 6
derived in a systematic manner from QCD, these formulations were also used in many
first principles lattice spectroscopic studies [45, 46, 47, 48]. Various forms of relativistic
potentials were also used in many studies. For example, in Ref. [49] a relativized quark
model with simple Breit-Fermi potential, combined with the spin-orbit interactions after
accounting for the corrections from Thomas precession, was considered. Ref. [50] uses a
relativistic potential for baryons constructed from a hyper-spherical expansion. Another
approach was to extract the spectrum by solving the three body problem of baryons
based on Bethe-Salpeter equations using relativistically covariant quark model in the in-
stantaneous approximation [51]. In Ref. [52], the relativistic three quark equations were
derived using dispersion relation techniques and were solved so as to extract the masses
of charm baryons with orbital excitations. In Ref. [53] a constituent quark model with a
full relativistic form for the light quarks and a velocity expanded relativistic potential for
the heavy quarks up to second order was used. Other calculations were also performed
based on QCD sum rules [54, 55], inspired by their success in explaining the low lying
heavy quarkonium systems. Masses and mass relations among the heavy hadrons were
studied assuming the quasi-linear Regge phenomenology ansatz for the heavy meson and
heavy baryon spectra [56]. Although predictions from all these models for the low lying
spectra are satisfactory, none of these models possess all the features of the full theory
of QCD and many of them also does not possess the full relativistic nature. Further,
it is questionable about the validity of the predictions for the extended baryons, where
the dynamics could include features beyond the approximation made in these models.
Hence, in the absence of any experimental discovery the only way to test these model-
dependent calculations is to compare these with the results from first principles lattice
QCD calculations.
1.2.2 Lattice spectroscopy
QCD, like other quantum field theories, needs to be regularized. Discretizing the space-
time serves as an intutive non-perturbative regularization scheme for the theory by intro-
ducing a UV cut off through the lattice spacing, ‘a’. A gauge invariant regularization of
QCD is possible this way, which is often referred to as lattice QCD. Such a regularization
combined with putting the theory on a finite sized box reduces the theory with infinite
degrees of freedom to a theory with finite number of degrees of freedom and thus allowing
to perform numerical computations of QCD. Expectation values of the QCD observables
can be expressed as finite dimensional path integrals, which can directly be evaluated
Introduction 7
with controlled systematics using Monte Carlo methods. At present, lattice QCD is the
only ab-initio approach to access the non-perturbative aspects of QCD, that includes the
QCD spectrum.
A quantitative description of the QCD spectra, particularly the charm baryons, using
rigorous ab-initio computations like lattice QCD is important for a number of reasons.
Firstly, many of the estimates of physical states from these calculations will be predictions
and thus can naturally provide crucial inputs to the future experiments. Secondly, lattice
results can provide guidance in identifying unknown quantum numbers (spin-parity) of
the discovered states, based on what one expects from QCD. Moreover, it will be interest-
ing to compare the low lying spectra of charm baryons computed from a first principles
method to those obtained from potential models, which have been very successful for
charmonia. It is expected that more information about interactions between multiple
charm quarks and charm quark and light quark can be obtained by computing the ex-
cited state spectra of charm baryons, including in particular the spin-dependent energy
splittings.
Lattice computations of hadron masses proceed through the calculations of the Eu-
clidean two point correlation functions, followed by performing non-linear exponential fits
to the long distance tail of these two point correlators, where only the ground state con-
tribution dominates. Lattice computations of light hadron masses are well documented
in the literature. Lattice studies with quenched approximation, which assume no quark
loops in the theory, were found to give estimates that are accurate with respect to the
observed values at ∼10% level [57, 58]. Light hadron spectroscopy calculations performed
in the preceding couple of years with dynamical configurations, that included the quark
loop effects from the light and strange flavors [59, 60], obtained an improvement in the
estimates and the results were observed to be accurate with the physical values at 3%
level of the statistical and systematic errors. Other dynamical calculations were also
performed, which used the fermion discretizations like O(a) improved Wilson action [61],
domain wall fermion action [62, 63] and twisted mass fermions [64]. But all of these
calculations compromised with one or other systematics involved. A notable work is by
the BMW collaboration [65], where they performed a precise determination of the masses
of the light hadron ground states, with controlled systematics that included the effects
of light and strange sea quark effects, the chiral extrapolations, the infinite volume limit
and the continuum limit. Their results were in excellent agreement with the experimental
observations.
The study of heavy hadron spectroscopy using lattice QCD, mainly for the ground
Introduction 8
state mesons, is quite mature. From early days of lattice QCD, impressive results were
obtained for spectra as well as for decay constants of heavy mesons, for example, in
Refs. [66, 67, 68, 69, 70, 71] and in many other works. Charmonium and bottomonium
mass splittings, particularly the 1S hyperfine splittings, were studied in detail in differ-
ent works using staggered quarks Asqtad improvement [72], Highly Improved Staggered
Quarks (HISQ) [73, 74, 75, 76, 77]. Notably the estimates from Refs. [76, 77] included
the quark loop effects up to the charm quark mass, various systematics including chiral,
infinite volume and continuum extrapolations. Their results include a set of predictions
along with a set of post-dictions, which are found to be in good agreement with the
observed values. Considering the heavy mass of the charm quarks, Ref. [78, 79] used
relativistic heavy quark formulation for the charm quarks, to determine the low lying
spectra of charmonium and charm mesons.
For charm baryons, early calculations include those using quenched configurations
[80, 81, 82]. Recently, the interest in lattice charm baryon spectroscopy has revived with
quite a few calculations being reported. Ref. [83] used relativistic Fermilab formulation
for the valence heavy quark and domain wall fermions for the valence light and strange
quarks on Nf = 2 + 1 gauge configurations using improved Kogut-Susskind sea quarks to
study the spin-1/2 singly and doubly charm baryons. Nf = 2+1+1 gauge configurations
using HISQ action for sea quarks was utilized by Ref. [84], to study the ground state
charm baryons, using relativistic heavy quark action for charm quarks and clover-Wilson
fermions for light and strange quarks. The masses of low lying charm baryons are evalu-
ated using two degenerate flavors of twisted mass sea quarks in Ref. [85]. There are a few
dynamical calculations using clover-Wilson fermions [86, 87] and relativistic heavy quark
action for charm quarks [88] aimed at spectroscopy of the low lying charm hadrons and
improving the discretization errors using different discretization schemes for the heavy
quarks [87].
1.2.3 Excited state spectroscopy
In spite of all these motivating results for low lying hadrons from lattice QCD, estimation
of excited state spectrum remained a major challenge. Lattice results are available only
for a few excited states [89, 90, 91, 92, 93, 94, 95] using usual local operators which
fails to give higher radial and orbital excitations. This deficiency is mainly because the
excited state contributions decay faster than those from the ground state. Hence the
conventional fitting procedures fail to produce reliable estimate as the signals for the
Introduction 9
excited states are swamped by those from the low lying states. One solution to study
the excited states is to use very fine temporal resolution and scan early times so as to
get stronger signals from the excited states. This can be achieved by using anisotropic
lattices, where one uses a finer temporal lattice spacing in comparison with the spatial
lattice spacing. Further, one can use a large set of operators that allow strong overlap with
the excited states. Use of novel smearing technique, called distillation [96], aid in efficient
computing of correlation functions between such large number of interpolating operators.
Development of distillation and the proper utilization of fitting techniques, such as the
generalized eigenvalue method [97, 98, 99], along with the derivative based operator
construction formalism [100, 101], boosted the hadron spectroscopic studies, facilitating
the determination of the excited state spectra [101, 102, 103, 104, 105, 106, 107, 108].
A significant part of this thesis consists of spectroscopy of charm baryons using two
complementary non-perturbative approaches. The first, which constitutes the major
part, employs these novel techniques mentioned above, enabling us to extract a tower of
spin identified excited states for all charm baryons. In this work, we follow a well de-
fined procedure developed and employed extensively for extracting excited states for light
mesons [103, 104, 105], mesons containing charm quarks [106, 107], and light and strange
baryons [101, 102, 108]. The procedure uses anisotropic lattice configurations with the
light and strange quark dynamics included [109, 110]. The anisotropic discretization
helps us obtain better resolution of the correlation functions, which is very helpful for
extraction of excited states. The anisotropic gauge configurations utilized here are gen-
erated with Nf = 2 + 1 flavor clover improved Wilson fermions [109, 110]. The valence
quarks, including the charm quark, are realized using clover action, which are O(am)
improved at tree level in tadpole-improved perturbation theory [111]. Furthermore, we
follow the derivative based operator construction formalism as described in Ref. [101].
This procedure proceeds in two steps : First we construct a large set of baryonic opera-
tors in the continuum. These operators transform as irreducible representations (irreps)
of SU(3)F symmetry for flavor, SU(4)S symmetry for Dirac spins of quarks and O(3)
symmetry for orbital angular momenta, corresponding to a SU(12)⊗O(3) symmetry. If
instead of the Dirac spinors with four components, Pauli spinors that are formed from
purely the upper two components of the Dirac spinor in Dirac-Pauli representation are
used, one would have obtained only an SU(6)⊗O(3) symmetry, where the SU(6) is built
up from the sub-groups according to SU(6) = SU(3)F ⊗ SU(2)S. The O(3) corresponds
to the symmetry in the spatial part. These continuum operators are then subduced to
various lattice irreducible representations to obtain lattice operators [101]. Finally, using
Introduction 10
the novel technique called “distillation” [96], correlation functions for these operators
are computed and the variational method is utilized to extract excited energies of charm
baryons with one or more valence charm quark content as well as to reliably determine
the spins of these states up to spin 7/2. We also computed several energy splittings, for
example, hyperfine as well as spin-orbit splittings between various states. Being first of
its kind in the excited charm baryon spectra calculation, this study serves as a founda-
tion for many follow up studies, which may help us address many of the challenges in the
baryon spectroscopy. The results from these studies were reported in Ref. [112, 113, 114].
Like any other lattice study, this work also possess certain systematics. The finite
lattice spacing introduces a UV cut-off and the finite box size brings in an IR cut-
off. It is very crucial that any physics extracted from these studies remains unaffected
by these cut-offs. In order to have good control over the ma errors, we used the tree
level tadpole improved clover action with O(mat) and O(mas) errors removed. We have
not addressed other higher order terms, as the temporal lattice spacing is quite small.
We also do not make a quantitative study of O(m2a2s) errors. However, to study the
charm physics, it is essential to have an action with discretization errors as small as
possible. Further, one needs to perform lattice calculations with multiple lattice spacings
on multiple lattice sizes; this is beyond the scope of this work. Furthermore, the light
quark mass on the lattices we use is such that the pion mass is ∼ 391 MeV. Along with the
approach to the thermodynamic limit, one also needs to perform a chiral extrapolation
due to the unphysical pion mass in these calculations, which again is beyond the scope
of this work. Another caveat in the above study is that the clover action does not have
chiral symmetry at finite lattice spacing, i.e., we have not addressed the effects of chiral
symmetry on observables. Though this may not affect the triply charm baryon spectra,
charm baryons with valence light quark content could have some effects. Furthermore, it
is also to be noted that we have not used any multi-hadron operators, which are expected
to give significant implications in the extracted spectra. Hence, one also need to perform
calculations considering multi-hadron operators. Keeping these caveats in mind in future
it will be worthwhile to pursue same calculation by addressing all these systematics to
make more precise quantitative predictions which can then confront experimental results.
1.2.4 Spectroscopy with chiral fermions
So as to complement the above calculation on spectroscopy of excited charm baryons,
we perform a second non-perturbative calculation of the low lying charm hadron spectra
Introduction 11
with highly controlled systematics. The main advantage of performing these calculations,
which cover the second part of the work on spectroscopy of charm baryons, using the
mixed action approach is the better control over the systematic errors that appear in the
estimates. Mixed action approaches have been studied by many groups such as DWF
valence on staggered fermion sea [63, 115, 116, 117], overlap valence on DWF sea [118],
overlap valence on clover sea [119], and overlap valence on twisted fermion sea [120].
This calculation uses overlap valence quarks in the background of a large set of dynami-
cal configurations generated with the one-loop, tadpole improved [111] Symanzik gauge
action and the highly improved staggered quark (HISQ) fermion action [121, 122]. Taste
violations, which were a generic problem with the unimproved staggered fermion action,
were found to be small in the HISQ formalism [121]. The overlap formalism has exact
chiral symmetry [123, 124] on the lattice and is automatically O(a) improved. By adopt-
ing such a mixed action approach, one can simultaneously get the advantage of having
a large set of configurations with very small discretization errors as well as small taste
breaking effects and the unbroken chiral symmetry as well as the low quark mass limit of
overlap fermions. The overlap action also has some desirable features computationally,
such as the adaptation of multi mass algorithms [125], which allows us to perform quark
propagator construction for multiple quark masses very efficiently. One also gets the
advantage of simulating both light, strange as well as heavy fermions on the same lattice
formalism with chiral fermions having no O(a) errors. With a large set of quark masses
being studied, we plan to perform a chiral extrapolation so as to get the estimates at
the physical pion mass. With the above formulation we have calculated the ground state
spectra of various charm hadrons and have made a comparative study between these two
projects along with other results from literature where available. Agreement between
these results from different approaches gives confidence in our estimates. The results
from these studies were reported in Ref. [126, 127].
1.3 Hadrons at finite temperature
Following the zero temperature calculations involving the spectroscopy of the low lying
hadrons, we are further interested in understanding how these physical states behave at
finite temperature. QCD at finite temperature proceeds through a crossover transition
between two phases : deconfined chirally symmetric phase at high temperatures to a
confined chiral symmetry broken phase at low temperatures. It has been observed from
many lattice calculations that this proceeds through a cross-over transition during this
Introduction 12
change of phase. While in a pure gluonic field theory, this change of phase proceeds
through a first order transition. The non-perturbative effects that dominate the low
temperature regime of QCD, contributes at a significant level even at finite temperatures.
In this work, we are interested in understanding how the degrees of freedom, in particular
those with nucleon quantum numbers, that exists in the zero temperature QCD gets
modified with increasing temperature.
In order to understand the thermal effects on the zero temperature degrees of free-
dom, we study the static correlation lengths that exist in a pure SU(3) gauge theory
above and immediately below the deconfinement transition temperature, Tc. Such static
correlations that persist in the equilibrium thermodynamic system can be studied by
introducing static probes into the equilibrium plasma and measuring the response of the
medium. This response depends on the quantum numbers carried by the probes; so one
can classify static correlators as glueball-like, meson-like and baryon-like probes with the
usual quantum numbers of these quantities corrected for the fact that the static spatial
symmetries are different from the Poincare group.
Meson-like screening masses have been studied in QCD in great detail [128, 129, 130,
131, 132, 133, 134, 135]. Baryons at finite temperature has not been studied in detail
in recent past and a few notable works were in the late 1980’s [128, 133]. Moreover, a
detailed study of the baryon screening masses in the low temperature phase also has not
been performed yet. In this work, we perform simulations of pure gauge theory for three
different temperatures across the transition temperature and study screening correlators
for mesonic and nucleonic resonances with clover fermions. The main emphasis, in this
work, has been given to the nucleon channels, which are expected to provide important
inputs to the study of baryon number fluctuations and thus to the experimental search
for the critical point of QCD. The results from these studies were reported in Ref. [136].
1.4 Summary
In this chapter, we give a detailed discussion about the phenomenological and theoretical
background in the spectroscopy calculations of baryons from lattice QCD. We begin with
briefing the major questions that are expected to be answered with the experimental
progress in the baryon spectroscopy, in Section 1.1. While in Section 1.2, we discuss
the main part of the thesis work, that includes the charm baryon spectroscopy. We
begin by emphasizing the experimental status and the theoretical developments from first
principles lattice QCD calculations and quark model calculations of the low lying heavy
Introduction 13
hadron spectrum. Following these discussions, we discuss our two main complementary
non-perturbative calculations, which cover most part of this thesis, of the heavy baryon
spectrum. The first calculation focuses on excited charm baryon spectroscopy using clover
fermions (Section 1.2.3), while the second calculation performs a spectroscopy of the low
lying heavy hadron states, with improved control over the systematics related the lattice
formulation of QCD (Section 1.2.4).
In the last section, we discuss our studies on hadron screening masses to study the
the nature of the strongly interacting matter at high temperatures. Main emphasis
was given for the study of baryonic screening lengths above and immediately below the
deconfinement transition temperature.
Chapter 2
Quantum ChromoDynamics on
lattice
Quantum ChromoDynamics (QCD) is the established fundamental theory of strong in-
teractions that governs the dynamics in a system of quarks and gluons. Along with the
weak and the electromagnetic gauge theories, QCD, which is an SU(3) gauge theory,
constitutes the Standard Model of particle physics. Perturbation theory, which relies on
the smallness of the coupling strength, provided an excellent tool to understand high
energy regime of QCD [7, 8]. However, first principles studies of QCD at the low energy
regime that includes spectroscopy, remained a highly non-trivial task.
The lattice formulation of QCD makes calculations of the QCD observables in the low
energy regime possible. Introduced by Kenneth G. Wilson in 1974 [137], lattice QCD is a
first principle non-perturbative regularization scheme that proceeds by putting the QCD
action on a discretized Euclidean space-time array. On a finite lattice, with only finite
number of degrees of freedom, it serves as a promising tool to study QCD using numerical
techniques. This formalism is widely used in determining various QCD observables. The
discretization introduces a UV cut-off, while the finite size of the lattice gives the system
an IR cut-off. All the infinite dimensional path integrals representing physical observables
become finite dimensional numerical integrals similar to those in statistical mechanics.
One can then use the Markov-chain Monte Carlo methods, which have been widely in
use for studies in statistical mechanics, to determine the observables in QCD.
This chapter is dedicated to a brief introduction to the theory of QCD on lattice.
Section 2.1 discusses the continuum formulation of QCD, while Section 2.2 briefs how
the QCD action is realized on the lattice and Section 2.3 details how simulations and
14
Lattice QCD 15
spectroscopy calculations are made using this lattice formulation of QCD.
2.1 QCD in the continuum
The QCD action in a four dimensional Euclidean space-time, with temporal extent at 0
and T , can be written as [138, 139, 140]
SE
QCD[ψ, ψ, A] = SF [ψ, ψ, A] + SG[A], (2.1)
where
SF [ψ, ψ, A] =
Nf∑f=1
∫ T0
dτ
∫d3x ψ(f)
α c(x)(γµ)αβ(δcd∂µ + iAµ(x)cd) +m(f)δαβδcd ψ(f)β d(x)
(2.2)
is the Fermion action and
SG[A] =1
2g2
∫d4x Tr[Fµν(x)Fµν(x)] (2.3)
is the gluon action with the bare gauge coupling, g.
Here ψ(f)(x) is the Fermion field corresponding to a quark flavor, f, with mass m(f)
and γµ is the Euclidean Dirac gamma matrix. (δcd∂µ + iAµ(x)cd) is the gauge co-
variant derivative with Aµ being the gauge field, which can be decomposed as Aµ(x) =∑i λ
iAiµ(x). Here λi’s are the generators of the SU(3) gauge group. Fµν(x) is the field
strength tensor, which can be expressed as Fµν(x) =∑8
i=1 F(i)µν (x)λi, where F
(i)µν can be
written in terms of the vector potential as
F (i)µν = ∂µA
(i)ν (x)− ∂νA(i)
µ (x)− fijkA(j)µ (x)A(k)
ν (x), (2.4)
where fijk’s are the structure constants.
This theory can be quantized using the Euclidean path integral formalism, in which
the partition function for this system can be written as,
ZT = Tr(e−T H) =
∫D[ψ, ψ, A]e
−SEQCD
[ψ,ψ,A]. (2.5)
The expression in the center represents the partition function in the Hamiltonian oper-
ator formalism, where the self-adjoint operator, H, is the Hamiltonian operator of the
Lattice QCD 16
system that measures the energy in the system and also governs the time evolution. The
expression on the right is the partition function in the path integral formalism. The
expectation value of an operator O is given by
〈O〉 =1
ZTTr(e−T HO) =
1
ZT
∫D[ψ, ψ, A]e−S
EQCD[ψ,ψ,A]O[ψ, ψ, A]. (2.6)
The usage ‘partition function’ for ZT appears very natural, once we establish the
structural equivalence between quantum field theory and statistical mechanics. For a
canonical ensemble of spins (s), the expectation value of an observable O is given by
〈O〉 =1
Z
∑s
e−βH[s]O[s]. (2.7)
where Z =∑s e
−βH[s] is the partition function, 1/β is the temperature in units of kB
and H[s] is the Hamiltonian of the system. The similarity between eq. (2.6) and eq. (2.7)
is very much evident. The Boltzmann factor, e−βH[s], is replaced by the weight factor,
e−SE
QCD , and the summation over all spin configurations by the integral over the fields
ψ, ψ and Aµ at all space-time points. The structural equivalence allows one to apply
the numerical methods, like Monte Carlo techniques, originally developed in statistical
mechanics, to quantum field theories.
2.2 Lattice formulation of QCD
Lattice QCD proceeds by the discretization of the space-time and thus regularizing the
theory from UV divergences by introducing a cut-off. The theory is put on a finite sized
box, so as to perform numerical computations, and this introduces an IR cut-off. The
next step is to define the QCD action on lattice such that it is explicitly gauge invariant
at any lattice spacing ‘a’. Further, any such definition for the QCD action on the lattice
should approach the continuum form in the limit, a→ 0.
The quark fields are placed on the lattice sites, followed by the discretization of the
derivative terms. Naive discretization of the derivative terms in the action,
∆µψ(n) =1
2a(ψ(n+ µ)− ψ(n− µ)), (2.8)
introduces terms involving fields at different space-time points resulting in gauge de-
pendent expressions. For example, consider the term ψ(n)ψ(n + µ) : under a gauge
Lattice QCD 17
transformation,
ψ(n)→ ψ′(n)→ Ωµ(n)ψ(n), ψ(n)→ ψ′(n)→ ψ(n)Ω†µ(n)
and ψ(n)ψ(n+ µ)→ ψ(n)Ω†µ(n)Ωµ(n+ µ)ψ(n+ µ)
.
Uµ(n) is a new field living on the links between two lattice sites that is introduced so
as to construct gauge invariant action. Then
ψ(n)Uµ(n)ψ(n+ µ)→ ψ′(n)U ′µ(n)ψ′(n+ µ) = ψ(n)Ω†µ(n)U ′µ(n)Ωµ(n+ µ)ψ(n+ µ)
is gauge invariant, given Uµ(n) transforms as
Uµ(n)→ U ′µ(n) = Ωµ(n)Uµ(n)Ω†µ(n+ µ).
Objects with such transformation properties in the continuum are the path ordered
(P ) exponential integral (eq. (2.9)) of the gauge field (Aµ) along a curve (C) connecting
two points, x and y.
U(x, y) = P exp (
∫C
igAds). (2.9)
Based on the transformation properties, Uµ(n)s are interpreted as the lattice version of
the gauge transporters in the continuum and hence the lattice gauge fields are introduced
as
Uµ(n) = exp (iaAµ(n)). (2.10)
Since Uµ(n) lives on the links connecting the adjacent sites, they are called as link vari-
ables. Thus, all the quark fields ψ and ψ live on the lattice sites, while the gauge fields
live on the links between adjacent sites.
2.2.1 Gauge action on lattice
As mentioned above, the main requirement of any lattice definition of QCD action is
that they should explicitly be gauge invariant and should have proper continuum limit.
The gauge invariant objects that can be formed purely from link variables are various
path ordered closed loops. With this information it is very straight forward to define the
lattice form of gauge action in terms of these closed loops. The simplest definition of the
Lattice QCD 18
lattice gauge action, SG, is given by
SG[U ] =2
g2
∑n∈Λ
∑µ<ν
Re Tr(I − Uµν(n)), (2.11)
where Uµν(n), called the plaquette, are the smallest closed loops constructed by the
product of four link variables.
Uµν(n) = Uµ(n)Uν(n+ µ)U−µ(n+ µ+ ν)U−ν(n+ ν), (2.12)
where U−µ(n+ µ) = U †µ(n).
Expressing this gauge action in terms of the field strength tensor in the continuum
and the lattice spacing one gets
SG[U ] =a4
2g2
∑n∈Λ
∑µ,ν
Tr[Fµν(n)2] +O(a2). (2.13)
In general, it is advantageous to use a gauge action with improved discretization effects
defined by adding contributions from the large loops constructed out of six or more link
variables to the definition given in eq. (2.11). Such an improvement is called Symanzik
improvement. In our spectroscopy calculations, we use such improved gauge actions with
appropriate co-efficients, to correct for the O(a2) errors in eq. (2.11). The details are
discussed in the respective sections.
2.2.2 Fermion action on lattice
The continuum action for a free Fermion is given by
SF
=
∫d4x ψ(x)γµ∂µψ(x) +mψ(x)ψ(x), (2.14)
where the Dirac indices are suppressed for convenience. Defining the lattice derivatives
as symmetric differences, a naive discretization can be made as
Snaive
F=∑n,µ
ψnγµ∆µψn +m∑n
ψnψn =∑m,n
ψmDm,nψn, (2.15)
where ∆µψn = 12a
(ψn+µ − ψn−µ) and Dm,n = 12
∑µ γµδn+µ,m − δn−µ,m+mδm,n is called
the Dirac operator.
Lattice QCD 19
Since Fermions are anti-commuting in nature, they are described by the Grassmann
numbers. Using Matthews-Salam formula [141],
∫ N∏n=1
dψn dψn e−
∑Ni,j=1 ψiDi,jψj = det[D] (2.16)
the two point functions of the Fermionic fields or the Fermion propagator can be obtained
as
〈ψiψj〉 = −D−1ij . (2.17)
The lattice free propagator in the momentum space is given by
1
aD−1naive(p) =
−iγµsin(pµa) +ma∑µ sin
2(pµa) +m2a2. (2.18)
A free Fermion propagator in the continuum, D(p) = (iγµpµ + m)−1, has its pole
at pµ = (0, 0, 0, 0). While for the naive discretization of Fermions on the lattice, one
get additional poles at pµ = (π, 0, 0, 0), pµ = (0, π, 0, 0), ..., pµ = (π, π, π, π) (all the
corners in a Brillouin zone). Hence in the continuum, we get sixteen poles representing
sixteen Fermions instead of just one Fermion. This is the (in)famous Fermion doubling
problem. Associated with this is a no-go theorem, which has important consequences on
Fermion discretization schemes This no-go theorem was first formulated in the Ref. [142].
It states that describing the Fermionic fields on a lattice by a transitionally invariant
(D(x, y) = D(x − y)) Hermitian matrix that preserves the chiral symmetry and have
couplings that extend over a finite number of lattice spacings, one inevitably runs into
the problem of Fermion doubling.
Wilson Fermions
One solution to get away with Fermion doubling is to force the unwanted solutions to be
heavy in the continuum limit and thus retaining only one low energy solution. Wilson
Fermions involve such a solution, where one adds an irrelevant operator constructed of
two derivatives.
SW
= rψDWψ = − r
2a
∑n,µ
(ψn+µ − 2ψn + ψn−µ) ∼ −ar2ψD2ψ, (2.19)
Lattice QCD 20
where DW is called the Wilson term forces the doublers to be heavy. With this definition,
the momentum space propagator becomes
1
aD−1naive+W (p) =
1
aD−1w (p) =
−iγµsin(pµa) +ma+ r∑
µ(cos(pµa)− 1)∑µ sin
2(pµa) + ma+ r∑
µ(cos(pµa)− 1)2(2.20)
While the doubler modes are lifted to energies that are of the order of 1/a at any
non-zero r, the pole at pµ = (0, 0, 0, 0) remains unchanged and thus retaining only one
low energy solution.
The Wilson term is not chiral, due to the presence of a term proportional to the mass
term in it, that causes explicit breaking of chiral symmetry. Thus the chirality of the
Fermion action is sacrificed in this formulation according to the aforementioned no-go
theorem. Consequently, the Wilson action may not be able to capture the subtle effects
of the spontaneous chiral symmetry breaking. This explicit symmetry breaking due to
the Wilson term cannot be rectified by further adding irrelevant terms to the lattice
action and is inherent to the theory. Another unpleasant disadvantage with the Wilson
Fermions is that the discretization errors appears at O(a), unlike the case of naıve lattice
Fermions, where the discretization errors start at O(a2).
Symanzik improved Wilson Fermions
Symanzik improvement program offers a clean solution to avoid the leading cut off effects.
The basic idea is to use additional terms in the action with dimension, d > 4, and tune the
respective co-efficients such that all contributions of order O(ad−4) are removed. First
application of this methodology to remove the O(a) errors in Wilson Fermions is by
Sheikholeslami and Wohlert [143]. They used the counter terms up to O(a5). The only
effective term at O(a5) other than the Wilson term that contribute to this improvement
is
SSW = cSW ψσµνFµνψ, (2.21)
where Fµν , known as the clover term, is the imaginary part of the sum of the four
oriented loops of link variables as shown in Figure 2.1. As was mentioned for the Wilson
Fermions, the addition of irrelevant terms to the lattice action does not restore the
explicitly broken chiral symmetry. Thus the chiral symmetry is broken in this formulation
also, in accordance with the aforementioned no-go theorem.
For the tree level improvement, cSW = 1. A non-perturbative evaluation of the co-
efficient, cSW (g) has been performed by Luscher, et al. [144] and an empirical form for
Lattice QCD 21
Figure 2.1: The ‘clover’ term
cSW was obtained,
cSW =1− 0.656g2 − 0.152g4 − 0.054g6
1− 0.922g2. (2.22)
It is to be noted that this empirical form has been derived for isotropic lattices. While
for the calculations using anisotropic lattices, we used the tree-level tadpole improved
values for the spatial and temporal improvement co-efficients.
Staggered Fermions
The doublers contribute from each corner of the Brillouin zone. Staggered formulation
ameliorates the doubling problem by reducing the size of the Brillouin zone from 1/a
to 1/2a. Effectively, the doubler Fermion degrees of freedom are distributed over a
hypercube of lattice spacing 2a on the lattice. This method was originally suggested by
Kogut and Susskind [145] and hence it is also known as Kogut-Susskind Fermions. The
staggered Fermion action is given by
SksF =1
2a
∑n,µ
χnαµ(n)χn+µ − χn−µ+m∑n
χnχn, (2.23)
where αµ(n) = (−1)n0+n1+...+nµ−1 .
χ and χ are the single component spinors. Because of the alternating nature of the
sign of αµ(n), the natural unit for the staggered Fermion field is the 24 hypercube. The
sixteen hypercube components are formed by four ‘taste’s of four ‘Dirac component’s.
The taste degree of freedom corresponds to the residual doubler degree of freedom. On
a finite lattice and for massless Fermions this action is invariant under
ψ → eiθψ & ψ → eiβ(γ5⊗τ5)ψ,
Lattice QCD 22
where τ5 is the γ5 matrix acting on the taste space. Thus the action has a remnant
U(1)⊗ U(1) chiral symmetry and this makes it useful in the study related to the chiral
symmetry. With the partially preserved chiral symmetry, the assumption in the no-go
theorem that is relaxed in this Fermion formulation is the remnant doubler degrees of
freedom (taste) contributing to the low energy physics.
Associating different taste with different physical flavors were considered in many
older works [146], but it was proved to be not a fruitful idea, because it is not possible
to break the flavor symmetry by introducing different masses for different tastes. One
alternative is to introduce new species of staggered field with its own mass for each flavor.
But this comes with a penalty of having additional multiplicity : each flavor comes with
its own four tastes. In common practice one uses one staggered Fermion for each flavor
and take the fourth root of the Fermion determinant [146].
Overlap Fermions
One way to realize chiral Fermion on lattice is through overlap formulation invented by
Narayanan and Neuberger [123]. The overlap action is given by
Dov = 1 + γ5sgn(γ5Dw); sgn(γ5Dw) =γ5Dw√
γ5Dwγ5Dw
, (2.24)
where Dw is the Wilson Dirac Fermion operator and sgn denotes the matrix sign function.
The operator possess a form of chiral symmetry on the lattice [147] :
δψ = iαγ5(1− a/2Dov)ψ & δψ = iαψ(1− a/2Dov)γ5, (2.25)
that leaves the Fermion action invariant for any a 6= 0. In the continuum limit, this
transformation reduces to the continuum definition of chiral transformation. Hence this
is interpreted as exact chiral symmetry at a 6= 0 and thus the overlap operator is bet-
ter suited for study of problems related to chiral symmetry. However, the presence of
the matrix sign function makes the overlap operator highly non-local and so it is very
expensive to implement it in numerical calculations. Thus the assumption of the local
nature of the couplings in the Fermion formulation is relaxed in accordance with the
no-go theorem.
One important symmetry of these lattice Dirac operators, that has very important
Lattice QCD 23
implications on numerical computations, is the γ5-hermiticity :
(γ5D)† = γ5D or γ5Dγ5 = D†. (2.26)
One important consequence of this property is that the eigenvalues of a γ5-hermitian
Dirac operator are either real or come in complex conjugate pairs. This implies that
the Fermion determinant, which is the product of all the eigenvalues, is real and hence
det[D] = det[D†].
Followed by these definitions for the Fermion action in the free field theory, one can
construct the Fermion action in the non-trivial gauge field theory by inclusion of the link
variables, which acts as gauge transporters and make the action gauge invariant. In our
calculations dealing with excited state spectroscopy, we use the O(a) improved clover
Fermion action on lattices with non-degenerate spatial and temporal lattice spacings
(anisotropic lattices). While in the studies employing the MILC configurations, the
dynamical simulations were carried out using a highly improved version of the staggered
quarks (HISQ) [121] and the spectroscopy calculations used overlap quarks. In the finite
temperature studies, the clover improved Wilson Fermions were used on isotropic lattices.
2.3 Numerical calculations using lattice QCD
In this section, we briefly discuss how simulations and spectroscopy calculations are
performed numerically using the lattice formulation of the theory of QCD.
2.3.1 Simulations using lattice QCD
Following the discretization of the space-time and the definition of the gauge and the
Fermion fields on the lattice, we move on to the discussion of how simulations are per-
formed on the lattice. The Fermion fields are represented by anti-commuting Gramss-
mann valued field variables so as to satisfy the Fermi statistics. Since there is no existing
realization of the Grassmann valued variables on a computer, one integrates out the
Fermion degrees of freedom and obtain the partition function as
Z =
∫D[U ] e−SG
Nf∏f=1
det[Df ] (2.27)
Lattice QCD 24
. Given eq. (2.26), for mass degenerate pair of quark flavors in dynamical simulations
with a γ5-hermitian Dirac operator, this weight factor (∏Nf
f=1 det[Df ]) will be positive
definite real number, which is a requirement for the application of Monte Carlo methods.
det[D] det[D] = det[D] det[D†] = det[DD†] ≥ 0. (2.28)
Monte Carlo techniques
Although the lattice formulation of the QCD in a finite sized box reduces the problem
of infinite degrees of freedom to a problem of finite degrees of freedom, the partition
function, Z, is still very large dimensional and naıve numerical integrations are not
possible even on small lattice within finite time. The structural equivalence (Section
2.1) of the quantum field theories and statistical mechanics comes to aid here. Monte
Carlo techniques, originally developed in the context of statistical mechanics, can be
utilized to find a representative sequence of statistically independent field configurations,
Ω, on which the observables can be measured such that
〈O(Ω)〉Ω '1
Z
∑i
e−SiO(Ωi). (2.29)
The underlying idea behind the Monte Carlo methods is as follows : Since the number
of configurations is very large, the weight factor for most of the configurations, Ωi, is very
small. Hence a simple set of randomly picked configurations from a uniform distribution
would result in a very bad convergence of eq. (2.29) and so highly inefficient. In Monte
Carlo technique, one performs an importance sampling, in which the configuration se-
quence is generated with a probability distribution P (Ω) equal to the weight factor in
eq. (2.27).
P (Ω) =1
Ze−SG
Nf∏f=1
det[Df ]. (2.30)
Markov process is a method to generate such a sequence of configurations that rep-
resent the whole ensemble. The idea is to pick a new configuration, Ω′, with a transition
probability, T (Ω,Ω′), which depends only on the current configuration, Ω, and Ω′. The
procedure is repeated, so that after some time of thermalisation, the equilibrium, which
satisfies eq. (2.29), is reached. The transition probability, T (Ω,Ω′) should posses two
important properties, so that it is a Markov process and hence guarantees the approach
to the equilibrium distribution after the initial period of thermalisation.
Lattice QCD 25
1. Strong ergodicity : For every Ω and Ω′,
TN(Ω,Ω′) > 0 for every Ω and Ω′.
In other words, every configuration should be reachable in finite number of updates,
N.
2. The transition probability, T (Ω,Ω′), should be a map of P (Ω) to itself.
P (Ω′) =
∫dΩP (Ω)T (Ω,Ω′) for every Ω′.
It can be shown that the 〈O(Ω)〉 evaluated on an ensemble generated from such a
Markov chain indeed approaches the ensemble average, 〈O〉, with a statistical uncertainty
of the order of 1√n. Here n is the number of configurations in the set, Ω. Simulations
for the finite temperature studies were performed using a combination of one heat bath
and three over-relaxation updates [140]. While the gauge configurations, that were used
in these spectrum calculations, were generated with the Rational Hybrid Monte Carlo
algorithm [148, 149, 150]. The details of these algorithm and the other related subtleties
related with the simulations are very huge that they themselves can cover the original
size of this document. Hence we omit those and refer the interested reader to the Refs.
[109, 122].
The major part of the computation resource in simulations goes into the evalua-
tion of the Fermion determinants. So as to reduce the computational intensity, sim-
ulations can also performed after putting these Fermion determinants to unity. This
approximation known as quenched approximation, is equivalent to neglecting all the cre-
ation/annihilation of the quark-anti-quark pairs in the vacuum. Though this is an acute
unphysical approximation, it has been surprisingly successful in many spectroscopy stud-
ies.
The finite temperature studies that are discussed in this thesis are based on the
quenched approximation, where there are no quark loops allowed in the gauge configura-
tions, while both of our works on charm baryon spectroscopy employs dynamical configu-
rations. The excited state spectroscopy studies employ Nf = 2 light and 1 strange quark
fields in the configuration generation, where the charm quark fields follow the quenched
approximation. While the MILC configurations used for spectroscopy calculations were
generated with Nf = 2 light, 1 strange and 1 charm quark fields in the lattice action.
Lattice QCD 26
2.3.2 Spectroscopy on the lattice
All numerical simulations of lattice QCD begins with the determination of the low lying
hadron masses. Since most of the masses are experimentally known to a high precision,
the comparison of these to the numerical estimates is an important check to the lattice for-
mulation of QCD. Other lattice QCD calculations, like matrix elements, thermodynamic
properties, also begin with the spectroscopy calculation of the low lying hadron masses.
In this section, we brief the theory behind conventional spectroscopy calculations.
Using the standard time evolution of a Hamiltonian system, a two point correlation
function can be expressed as
〈Oi(t)O†j(0)〉 =
1
ZTTr(e−THetHOie
−tHO†j), (2.31)
where ZT , which is the partition function, which ensures proper normalization of the
correlation function. In the limit of large T , one can express the zero-momentum two
point correlation function as a sum of contribution from the eigenstates at rest,
limT →∞〈Oi(t)O
†j(0)〉 =
∑n
〈0|Oi|n〉〈n|O†j |0〉2 ∆En
e−t∆En =∑n
Zni Z
n∗j
2 ∆Ene−t∆En , (2.32)
where ∆En = En − E0 is the energy difference of the eigenstate relative to the energy
of the vacuum and the Zni = 〈0|Oi|n〉 is referred to as the overlap factors, which carries
the information about the quantum numbers of the states they represent. In Section 3.5,
we will see these quantities are crucial in the determination of the spin of the extracted
states. Since the calculation will be performed on a finite lattice, the∑
n is over a finite
number of states. Assuming discrete spectrum in the low lying states, in the large t limit,
the lowest lying state corresponding to ∆E1 contributes significantly, while contribution
from all other higher lying states can be neglected.
In brief, the first step in the lattice spectroscopy calculations is the identification /
construction of the hadron interpolators 1, which are functionals of the lattice fields with
the quantum number of the state one is interested in. Once the interpolators are identified
/ constructed, one evaluates the two point Euclidean correlation functions between them.
Finally, the low lying spectrum can be extracted by performing non-linear fitting to the
exponential fall of the correlation functions at large t, where the contribution from the
excited sates can be neglected.
1Interpolator and operator refers to the same quantity and we use them here synonymously.
Lattice QCD 27
Consider an isovector meson operator, OM(x) = ψuΓψd(x), where Γ carries the spin
quantum numbers and the spatial description of the operator. The two point Euclidean
correlators for this operator is then given by
〈 OM(x) OM(y) 〉 = 〈 ψuΓψd(x) ψdΓψu(y) 〉
= − Tr Γ D−1d (x, y) Γ D−1
u (y, x) , (2.33)
where D−1f (x, y) = 〈ψf (x)ψf (y)〉. The negative sign appears due to the rearrangement
of the Grassmann valued Fermion fields. From the above expression once can see that
the expectation value factorizes with respect to the flavors, 〈 ... 〉 = 〈 ... 〉f1〈 ... 〉f2 .
Application of the Wick’s theorem for each individual flavors expresses the correlation
function as a product of the quark propagators as given in the second step of eq. (2.33).
This procedure is called as Fermion contraction.
It can be seen from the eq. (2.33), the quark propagators carry no information about
the quantum numbers in the interpolator and all the information about the quantum
numbers of the state of interest are embedded in Γ factors. Hence, the computation
of the quark propagators can be performed independently. More importantly, once all
these quark propagators are computed, one can construct correlation functions for any
interpolators of interest.
2.3.3 Quark propagators
To compute the hadron correlation functions, we will have to compute the quark propa-
gators in each configurations. However, computing quark propagators from all the sites
on the lattice to all the sites on the lattice (all-to-all) would not be feasible, due to the
computational intensiveness and the huge memory requirement. For example, if we con-
sider all-to-all propagators on the lattices that we use for the excited state spectroscopy
(163× 128), quark propagator from each configuration consumes approximately 633 Ter-
abytes of memory, which is enormously huge. The simplest solution, that is widely
followed, is to put quark source (ηi) in just one point in space-time (point-to-all propa-
gator) followed by construction of the quark propagators from them as the solutions of
the matrix equations
D xi = ηi, (2.34)
where D is the Dirac matrix. However, for operators that are non-local, one may need to
compute point-to-all propagators using quark sources at multiple space-time points. We
Lattice QCD 28
employ novel smearing techniques to circumvent these necessities, the details of which
are described in the Section 3.4.7.
In general, this system of linear equations (eq. (2.34)) are solved by iterative methods
such as Conjugate Gradient (CG) for symmetric positive definite matrices, the MINRES-
method for symmetric non-definite matrices or Bi-Conjugate Gradient (BiCG) method
for non-symmetric matrices. These details of the quark propagator generation are either
described in brief or referred to literature in the respective sections.
2.4 Summary
In this chapter, we briefly discuss the QCD in the continuum (Section 2.1) and how it is
formulated on a lattice, so as to perform numerical computations (Section 2.2 and 2.3).
In Section 2.2, we discuss the details about the standard discretization scheme for the
gauge fields and the various lattice realization of Fermionic fields along with the main
pros and cons in each of the realizations. In Section 2.3, we briefly discuss the basic idea
behind Monte Carlo simulations and the spectroscopy calculations using lattice QCD.
This chapter serves as a fundamental building block for various topics discussed in later
chapters.
Chapter 3
Excited state spectroscopy
As mentioned in the previous chapter, spectrum calculation proceeds by the construction
of the two point correlation function on the lattice, followed by fitting them with non-
linear fit forms and extracting the energy spectrum of QCD from the exponential fall of
these correlation functions at large Euclidean times. While the ground state energy can
be determined fairly straight forward from the correlations at large time, the extraction
of the excited state spectra is highly non-trivial. This is because the signals from the
excited state appears in the sub-leading exponentials and simple multi-dimensional fits
are usually unstable. In this section, we start by discussing the conventional methodology
in extraction of the ground state masses from the large time exponential fall-off in Section
3.1, followed by the challenges in excited state spectroscopy on the lattice in Section
3.2. Then we discuss the solutions of these challenges including the use of anisotropic
lattices, construction of large basis of carefully designed interpolators, and a good fitting
technique, which also aids in reliable spin identification in Section 3.3, 3.4 and 3.5.
3.1 Conventional spectroscopy from lattice
As discussed in the previous chapter, spectroscopy proceeds by constructing the two point
correlation functions between the operators with the desired quantum numbers. Based
on the spectral decomposition, the two point correlation function at large times behaves
as
C(t) ∼ Ae−Et, (3.1)
where E is the energy gap of the ground state from the vacuum. If the energy gap
between the ground state and the excitations are large enough, the ground state can
29
Excited state spectroscopy 30
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
2 4 6 8 10 12 14
Eff
ecti
ve m
ass
t/at
Figure 3.1: The plot of effective mass versus the time slice, showing the plateau in theeffective mass at large times. At short times, the contribution from the excited states isclearly visible from the approach to the plateau. The fit value is an estimate using thefit range 8-14.
be extracted reliably by fitting it with single exponential at large Euclidean times. The
non-negligible contamination of the excited states will be evident as the instability in the
fit range. This can also be viewed in terms a local fit estimate, known as effective mass
or local mass, which is defined as
at meff = ln(C(t)
C(t+ 1)). (3.2)
This quantity helps to visualize the decaying contribution from the excited states in the
correlation function as the approach to plateau in effective masses.
The final step in the extraction is to compare the fit estimates with the plateau in
the effective mass plot. This method is very efficient as the effective masses are very
much visually appealing than the correlation functions itself, as the correlation function
involves numbers that differ in magnitude by many orders. Thus, reliable fits can be
made in those ranges with negligible contribution from the higher lying states to the
Excited state spectroscopy 31
effective masses by ensuring a stable plateau in the fitted range of time slices. In Figure
3.1, we show a sample effective mass plot, where the best fit result (fit range 8-14)is also plot. One need to be careful with the fact that, if the covariance between the
time-slices are large then the good fit range may not pass through the effective masses.
3.2 Challenges in excited state heavy hadron spec-
troscopy
Though the basic procedure in all spectroscopy calculations on the lattice are similar and
fairly straight forward, the challenges in the excited state heavy hadron spectroscopy
are multiple fold. The first obvious difficulty is attributed to the heaviness of the quark
in the heavy hadrons. The discretization errors in a lattice fermion action and the
Fermionic observables are quoted in terms of the dimensionless quantity am, where m
is the quark mass. The typical lattice spacings that are used for the light hadron spec-
troscopy (O(0.1 fm)), cannot be used for the heavy quark systems, as they will provide
very large discretization errors to the estimates. For rapidly decreasing successive cor-
rections from the higher order terms in am, one requires am << 1, so that one can
neglect the contributions beyond some order of am. This requires one to use a lattice
with very fine spacing between the lattice points, so that am << 1. Further, the physical
states comprised of heavy quarks, being heavy, the correlations fall very rapidly with the
Euclidean time. Employing very fine lattices in the calculation becomes a requirement
due to this necessity for high temporal resolution at very early time slices.
The study of excited states also comes with various difficulties. Lepage analysis [151]
of fluctuations in hadron correlations shows that while signal-to-noise ratio (SNR) in
meson correlations is a constant with Euclidean time, the SNR in baryon correlations
falls exponentially. The argument is as follows. The signal for mesons and baryons can
be extracted from the respective correlations. While the fluctuations in such correlations
are given by C2, which goes as σ2M(τ) ∼ e−2 mπτ for mesons and σ2
B(τ) ∼ e−3 mπτ in
baryons. This is because each quark propagator is uncorrelated with any other in a
gauge field and does not know which physical state it is part of. Hence the variance in
the baryon correlations are dominated by the state with three pions, and hence σB(τ)
falls much more slowly than the signal as 3/2mπ < mB. Further, for the excited states,
Excited state spectroscopy 32
the scenario is much worse as 3/2mπ << m∗B and the SNR goes as
limτ→∞
CB(τ)
σB(τ)∝√Ncfg e
−(m∗B−3/2mπ)τ (3.3)
Employing a very fine lattice serves as a naive solution here also. We can extract the
energy of the resonances from very early time slices, where the signals from the desired
correlations may not have eclipsed by the noise. But using very fine lattice spacings
along all the four dimensions is extremely costly. So we follow a formalism, in which
the temporal lattice spacing, along which the correlations are measured, is much finer
compared to the spatial lattice spacings, while keeping the latter unchanged. This allows
one to reduce the huge computational necessity that will arise from reducing lattice
spacing in all four dimensions, while keeping the discretization errors under control. This
formalism, known as anisotropic formalism [109, 152, 153, 154, 155], is briefly discussed
in Section 3.3.
Another major difficulty is in the extraction of the excited state spectrum using
conventional fitting procedures. In principle, one can perform multi-parameter fits to get
the information about the higher excited states. But in general, the stability of the fits
degrades very fast with the addition of each successive exponentials to the fit form. Hence
it is desirable to have the correlation functions such that contributions from the higher
lying states to the state being studied are suppressed, while keeping contribution from the
state of interest intact. The solution to this problem is to perform a careful construction
of large basis of interpolating operators, which can overlap strongly with various physical
states. Marrying this with a novel smearing technique, called distillation, provides an
efficient method to compute the two point correlation functions for the large basis of
operators, in addition to suppressing the contribution from the higher lying states to
the state being studied. Details of this construction of the operators and the use of
distillation are discussed in detail in Section 3.4.
In addition to the computation of the correlation functions with improved signal from
the physical states, one also require to have a good analysis procedure in their extraction
along with precise determination of the respective quantum numbers. A reliable identifi-
cation of the spin-parity quantum numbers of a state is highly non-trivial on the lattice.
We use the variational fitting method so as to extract the spectrum of baryon states
from the matrix of correlation functions using large basis of interpolators. Section 3.5
discusses the details about the variational fitting method. We also discuss the procedure
we follow to identify the quantum number of the extracted state from these lattices with
Excited state spectroscopy 33
single lattice spacing using the overlap factors, Zni .
3.3 Anisotropic lattice
A possible solution for the issues related to the discretization errors as well as the rapid
decay of the temporal correlations is to increase the temporal resolution by adopting a
very fine lattice spacing along the Euclidean time direction, while keeping the spatial
lattice spacing same. This reduces the huge computational efforts that would have come
from reducing the spacing in all lattice directions [109, 152, 153, 154, 155]. In this work,
we adopt such an anisotropic dynamical lattice formulation to extract the highly excited
charm baryon spectra.
We use the anisotropic Nf = 2 + 1 flavor dynamical gauge configurations generated
by the Hadron Spectrum Collaboration (HSC)[109]. For the gauge sector, a Symanzik-
improved action was used. With tree level tadpole improved coefficients [111], the action
is given by
SξG[U ] =β
Ncγg
(∑x,s<s′
[5Pss′3u4
s
− Rss′
12u6s
]+∑x,s
[4Pst
3u2su
2t
− Rst
12u4su
2t
]), (3.4)
where P is the plaquette and R is the 2+1 rectangular Wilson loop. In order to preserve
the positive definiteness of the transfer matrix, length-two rectangle in time were omitted
from the action. The coupling g2 appears in the β = 2Ncg2 . The parameter γg is the
bare gauge anisotropy and us and ut are the spatial and temporal tadpole improvement
factors, dividing the spatial and temporal gauge links respectively. This action has leading
discretization errors O(α4s, a
2t , g
2a2s).
For fermions, we use an anisotropic Shekholeslami-Wohlert fermion (clover) action,
with tree level tadpole improvement and three dimensional stout link smeared gauge
fields. In terms of the dimensionless variables, m0 = m0at, ∇µ = a2µ∇µ, ∆µ = aµ∆µ, the
‘Wilson operator’ : Wµ = ∇µ − γµ2
∆µ and the clover term : Fµν = aµaνFµν , the fermion
action is given by
SξF [U, ψ, ψ] =∑
ψ(x)1
ut(utm0 + Wt +
1
γf
∑s
Ws
−1
2
[(γgγf
+1
γrg)
1
2utu2s
∑s
σstFst +1
γfu3s
∑s<s′
σss′Fss′
])ψ(x), (3.5)
Excited state spectroscopy 34
where γf is the bare fermion anisotropy, γrg is the renormalized gauge anisotropy, us
and ut are the spatial and temporal tadpole improvement factors associated with the
three dimensional stout smeared gauge fields and σµν = 12[γµ, γν ]. Finer details of the
formulation, the determination of the tadpole factors and the improvement co-efficients,
the tuning of the anisotropy coefficients and the algorithms employed in the simulation
of these lattices are detailed in the Ref. [109].
Lattice size atm` atms Ncfgs mK/mπ mπ/MeV atmΩ
163 × 128 −0.0840 −0.0743 96 1.39 391 0.2951(22)
Table 3.1: Properties of the gauge-field ensembles used. Ncfgs is the number of gauge-fieldconfigurations.
The important lattice parameters required for this work is summarized in the Table
3.1. The temporal lattice spacing, at, was determined using the Ω-baryon mass measured
on these ensembles [110]. This leads to a−1t = 5.67(4) GeV. On these lattices we find
mcat << 1, which validates the use of these lattices to study the charm physics. With
an anisotropy close to 3.5, as = 0.12 fm. This gives a spatial extent of about 1.9 fm,
which would be sufficient to study charm baryons, in particular doubly and triply charm
baryons.
The charm quarks were not included in the sea during generation of these lattices.
It is expected that the effects due to the absence of the dynamical charm quark loops
in this calculation will be small, as the disconnected diagrams are OZI suppressed. Liu,
et al. [106, 107] initiated the study of charm hadron spectroscopy using these lattices
and the construction of clover fermion perambulators (equivalent of quark propagator,
see Section 3.4.7). The action parameters for the charm quark, including the fermion
anisotropy and the temporal and the spatial clover co-efficients, are obtained by ensuring
that the mass of the ηc meson takes its physical value and its dispersion relation at low
momentum is relativistic. In their studies, the preliminary tests they made to estimate the
contribution of disconnected diagrams to the hyperfine splitting in charmonium showed
that this contribution is negligibly small.
3.4 Construction of baryon operators
After addressing the issue of discretization, the next challenge is to construct a large
basis of baryon interpolating operators, such that they project on to the desired states
Excited state spectroscopy 35
of interest. In this section, we discuss the construction of baryon interpolating operators
using up to two derivatives, such that they project up to J = 72
states with both even and
odd parities. We follow the prescription given in [101]. The details about the construction
of the operators used for the study also provides insight into the results obtained from
the numerical studies. Since our calculations are performed in a hyper-cubic lattice, all
the operators that we use should be classified according to the symmetry on the lattice,
rather than the full symmetry of the continuum space-time. As the lattice spacing is
made quite small, we expect to recover the symmetry of the continuum space-time.
Since we are only interested in spectroscopy, restricting ourselves to the zero-momentum
correlations, we require that our operators should transform according to the irreps of
the double-cover octahedral group, ODh . The methodology involves first constructing a
basis of baryon interpolating operators in the continuum with well-defined continuum
spin-parity quantum numbers. These continuum operators are then subduced/reduced
to the irreducible representations (irrep) of the double-cover octahedral group on the
lattice. The correlation functions, constructed out of these octahedral operators, are
expected to show rotational symmetry breaking artefacts in these calculations due to the
reduced symmetry on a lattice. However, we observe, as will be discussed in Section 3.6,
an effective rotational symmetry from the primary spin identification tests and that the
symmetry breaking effects are very small. In the next section, we will be discussing these
tests in detail with examples.
3.4.1 Continuum baryon interpolating operators
All hadrons are color singlet objects and thus they form totally anti-symmetric combi-
nations in the color indices of its constituents. Since baryons are fermions made of three
quarks, their interpolating operators excluding the color part should be totally symmetric
combinations of all the quark labels representing flavor, spin and the spatial structure.
The overall flavor-spin-spatial structure of a continuum baryon interpolating operator
with quantum numbers, JP , can be decomposed into a combination as
O[JP ] = [FΣF ⊗ SΣS ⊗DΣD ]JP
, (3.6)
where F , S and D are flavor, Dirac spin and spatial projection operators respectively and
the subscripts stands for the symmetry in the respective subspaces. For every baryon
operator, we must combine the symmetry projection operators such that the resulting
Excited state spectroscopy 36
baryon operator, excluding the color part, is overall symmetric. In the next subsection,
we discuss the general symmetry constructions possible for three objects, based on which
we construct the symmetry projection operators in individual subspaces.
3.4.2 General symmetry combinations
Before we start with construction of the continuum operators, let us look at the definite
symmetries for three objects that can be labeled by x, y and z, where the first object is
labeled by x, the second is labeled by y and third by z. There are four definite permutation
symmetry combinations: Symmetric (S), Mixed-Symmetric (MS), Mixed-Anti-symmetric
(MA) and Antisymmetric (A). They are as follows
xyzS
= NS
[ xyz + yxz + zyx+ yzx+ xzy + zxy ]
xyzMS
= NMS
[ xyz + yxz + zyx+ yzx− 2xzy − 2zxy ]
xyzMA
= NMA
[ xyz − yxz + zyx− yzx ]
xyzA
= NA
[ − xyz + yxz + zyx− yzx+ xzy − zxy ] (3.7)
The objects x, y and z could be identical as well, depending on which the normal-
ization constants NS, N
MS, N
MAand N
Achange. For example, when all of them are
different, NS
= 1√6, and when all of them are the same, N
S= 1. The convention used for
mixed symmetries is the same as used in [101], where the MS and MA are according to
whether the first two labels are symmetric or antisymmetric.
As described in eq. (3.6), baryon operators are constructed from direct products of the
different projection operators corresponding to the flavor, spin and the spatial derivative
labels. Each of these labels can be arranged according to the symmetry combinations
above, following which they have to be combined with each other in order to make an
overall symmetric object. The general rules for carrying out direct product between the
symmetry combinations of independent sets to make different overall symmetries are as
follows [101] :
1S2
S= 1, 2
S, 1
S2
MS= 1, 2
MS,
1S2
MA= 1, 2
MA, 1
S2
A= 1, 2
A,
1A2
S= 1, 2
A, 1
A2
MS= 1, 2
MA,
−1A2
MA= 1, 2
MS, 1
A2
A= 1, 2
S,
Excited state spectroscopy 37
1√2
(+1MS2
MS+ 1
MA2
MA) = 1, 2
S,
1√2
(−1MS2
MS+ 1
MA2
MA) = 1, 2
MS,
1√2
(+1MS2
MA+ 1
MA2
MS) = 1, 2
MA,
1√2
(−1MS2
MA+ 1
MA2
MS) = 1, 2
A, (3.8)
3.4.3 Flavor symmetry structures
(a)(b)
(c)
Figure 3.2: The SU(4) multiplets of baryons containing u, d, s and c flavors [16]. (a) 20M -plet containing the light-strange SU(3) octet, (b) 20S-plet containing the light-strangeSU(3) decuplet and (c) 4A-plet containing the light-strange SU(3) singlet
Figure 3.2 shows the SU(4)F - classification of the baryons containing the u, d, s and c
flavor quarks. The lowest plane in all these diagrams represent the octet, decuplet and the
singlet in the light-strange SU(3)F symmetric flavor sector. With the approximation that
mu ∼ md (the theory has an approximate isospin symmetry) and ms being only an order
of magnitude heavier, we see in nature that this multiplet structure is nearly followed.
But due to very heavy mass of the charm quark, we believe that there could be very strong
mixing between various multiplets in the physical states. Though this symmetry is greatly
broken to the large non-degeneracy in the masses, these provide a good bookkeeping
systematics for the large set of baryons that are allowed based on quark models. The
layers, except the lowest, constitutes of the charm baryons. With only three different
flavors simultaneously possible, the flavor wave functions of the charm baryon operators
can be constructed to be members of SU(3)F multiplets as 3 ⊗ 3 ⊗ 3 = 10S ⊕ 8MS ⊕
Excited state spectroscopy 38
8MA⊕1A. These represent a subset of states from the larger group of SU(4)F which has a
decomposition based on symmetry considerations as 4⊗4⊗4 = 20S⊕20MS⊕20MA⊕4A.
Table 3.2 contains the details of the flavor symmetry constructions possible for various
charm baryons.
20MI Iz S FMS FMA
Λ+c 0 0 0 1√
2(|cud〉MS − |udc〉MS) 1√
2(|cud〉MA − |udc〉MA)
Σ++c 1 +1 0 |uuc〉MS |uuc〉MA
Σ+c 1 0 0 |ucd〉MS |ucd〉MA
Σ0c 1 −1 0 |ddc〉MS |ddc〉MA
Ξ′+c
12 +1
2 −1 |ucs〉MS |ucs〉MA
Ξ′0c
12 −1
2 −1 |dcs〉MS |dcs〉MA
Ξ+c
12 +1
2 −1 1√2(|cus〉MS − |usc〉MS) 1√
2(|cus〉MA − |usc〉MA)
Ξ0c
12 −1
2 −1 1√2(|cds〉MS − |dsc〉MS) 1√
2(|cds〉MA − |dsc〉MA)
Ω0c 0 0 −2 |scs〉MS |scs〉MA
Ξ++cc
12 +1
2 0 |ccu〉MS |ccu〉MA
Ξ+cc
12 −1
2 0 |ccd〉MS |ccd〉MA
Ω+cc 0 0 −1 |ccs〉MS |ccs〉MA
20SI Iz S FS
Σ++c 1 +1 0 |uuc〉S
Σ+c 1 0 0 |ucd〉S
Σ0c 1 −1 0 |ddc〉S
Ξ+c
12 +1
2 −1 |ucs〉SΞ0c
12 −1
2 −1 |dcs〉SΩ0c 0 0 −2 |ssc〉S
Ξ++cc
12 +1
2 0 |ccu〉SΞ+cc
12 −1
2 0 |ccd〉SΩ+cc 0 0 −1 |ccs〉S
Ω++ccc 0 0 0 |ccc〉S
4AI Iz S φA
Λ+c 0 0 0 |udc〉A
Ξ+c
12 +1
2 −1 |ucs〉AΞ0c
12 −1
2 −1 |dcs〉A
Table 3.2: Flavor symmetry structures for charm baryons. I in the first column standsfor the isospin, Iz in the second column is for the third component of isospin and S inthe third column is for the strangeness.
With only one kind of flavor available, the Ωccc baryon, like the ∆ and Ω baryons,
appears only in the S flavor construction. While all those baryons with two of the quarks
having identical flavor, which is different from that of the third quark, can appear in the
S, the MS and the MA flavor constructions. The doubly charm baryons, Σc and the Ωc
baryons falls in this category. While the Ξc baryon that have all the three quarks in three
Excited state spectroscopy 39
different flavor can appear in all the symmetry constructions. Λc baryon, for which all
three quark flavors are different, being an isospin singlet, can appear only with MS, MA
and A flavor constructions.
3.4.4 Dirac spin symmetries
The Dirac matrices in the Dirac-Pauli representation can be expressed as a direct product
of two SU(2) components : one for ordinary spin (s-spin) and the other for intrinsic
parity (ρ-spin), where the SU(2) components are generated by 2 × 2 Pauli matrices
corresponding to the two component Pauli spinor ( +− ) [100]. This representation of Dirac
matrices is convenient for constructing the symmetry combinations that exists in the
Dirac labels. Expressed in terms the SU(2)ρ ⊗ SU(2)σ matrices, γ4 = ρ3 ⊗ σ4, where
ρ3 = diag[1,−1] and σ4 = diag[1, 1]. Thus the Dirac index µ = 1, 2, 3, 4 is equivalent to a
two-dimensional superscript corresponding to ρ-spin (ρ = +1,−1) and a two-dimensional
subscript corresponding s-spin (σ = +1,−1), and the quark field may be expressed as
ψµ = qρσ. The mapping of Dirac indices (µ) to the ρ and s labels are as given in Table
3.3(a). We refer to the ρ value as ρ-parity because of its role in the parity transformations
(γ4 is the parity transformation operator).
Using a two component Pauli spinor, σ-spin, and the formula in eq. (3.7), one gets the
following symmetry combinations for a three quark system, with total spin 32, 1
2and 1
2for
the symmetric (S), mixed symmetric (MS) and mixed antisymmetric (MA) combinations
respectively.
S :∣∣3
2,+3
2
⟩=∣∣+ + +
⟩S;
∣∣32,+1
2
⟩=∣∣+ +−
⟩S;∣∣3
2,−1
2
⟩=∣∣+−−
⟩S;
∣∣32,−3
2
⟩=∣∣−−− ⟩
S;
MS :∣∣1
2,+1
2
⟩=∣∣+−+
⟩MS
;∣∣1
2,−1
2
⟩= −
∣∣−+−⟩MS
;
MA :∣∣1
2,+1
2
⟩=∣∣+−+
⟩MA
;∣∣1
2,−1
2
⟩= −
∣∣−+−⟩MA
. (3.9)
Only the operators formed by considering a two component Pauli spinor (ρ = +1), which
form the upper two components of the four component Dirac spinor (µ), appear in the
leading order in a velocity expansion of any baryon operator. The baryon operators
constructed purely from such a two component Pauli spinor is hence non-relativistic in
nature. Those based on the lower components of Dirac spinors are relativistic. A full
relativistic formulation requires consideration of the ρ = −1 sector as well, hence one
uses the full four component Dirac spinor instead of the two component Pauli spinor.
Excited state spectroscopy 40
The symmetry constructions for the ρ-spins also follow similarly. The direct product of
the ρ-spin and the σ-spin follows the rules as described in eq. (3.8), and yields sums of
three-quark terms in which each quark has a ρ and s label representing its Dirac index.
The construction of the symmetry combinations for full four component Dirac spinor
using the symmetry patterns from ρ-spin and s-spin is detailed in Table 3.3(b).
Dirac ρ s1 + +2 + −3 − +4 − −
(a)
D IR Emb S = ρ⊗ sS 1
21 1√
2
(+ |ρ〉MS|s〉MS + |ρ〉MA|s〉MA
)32
1, 2 |ρ〉S|s〉SMS 1
21, 2 |ρ〉S|s〉MS
3 1√2
(− |ρ〉MS|s〉MS + |ρ〉MA|s〉MA
)32
1 |ρ〉MS|s〉SMA 1
21, 2 |ρ〉S|s〉MA
3 1√2
(+ |ρ〉MS|s〉MA + |ρ〉MA|s〉MS
)32
1 |ρ〉MA|s〉SA 1
21 1√
2
(− |ρ〉MS|s〉MA + |ρ〉MA|s〉MS
)(b)
Table 3.3: (a) : Mapping of Dirac spin indices to ρ and s labels. (b) Table taken fromRef. [101] : Symmetries of Dirac spin states based on direct products of ρ-spin and s-spinstates for three quarks. |ρ〉Σρ and |s〉Σs refer to a ρ-spin and s-spin state with Σρ andΣs symmetries respectively. Direct products of the ρ-spin and s-spin states yield sumsof three-quark terms in which each quark has a ρ and s label. These two labels togetherdetermines each quark’s Dirac index according to (a).
From the above discussion, it is clear that using spatially local constructions the total
spin of baryon can have only two values, 1/2 or 3/2. The flavor and non-relativistic
spin can be combined in an approximate spin flavor SU(6) whereby the multiplets are
6⊗ 6⊗ 6 = 56S ⊕ 70M ⊕ 70M ⊕ 20A. These can again be decomposed into flavor SU(3)
multiplets as 56 = 410⊕ 28, 70 = 210⊕ 48⊕ 28⊕ 21, 20 = 28⊕ 41, where the superscript
(2S+1) gives the spin for each particle in the SU(3) multiplet. The wave functions of the
56-plet can be combined with local or non-local spatial projections to make the non-color
component of the wave function symmetric, while the wave functions of the 70 and 20-
plets require some non-local excitations of the spatial part to make the overall non-color
component of the wave function symmetric. With the four component Dirac spinors, the
actual construction of the operators corresponds to SU(12), which is built up from the
subgroups according to SU(12) = SU(3)F ⊗ SU(2)ρ ⊗ SU(2)s. The details of the local
operators constructed from the symmetries in the flavor and Dirac spin sector, forming
the SU(12) group, is tabulated in the Table 3.4.
Excited state spectroscopy 41
Σ SU(3)SU(2) Nnr Nr
S (10,4) 1 1 φSχS
(10,2) 1 φSχS
(8,4) 1 1√2(φMSχMS + φMAχMA)
(8,2) 1 2 1√2(φMSχMS + φMAχMA)
(1,2) 1 φAχA
M (10,4) 1 MS φSχMS MA φSχMA
(10,2) 1 2 φSχMS φSχMA
(8,4) 1 1 φMSχS φMAχS
(8,4) 1 1√2(−φMSχMS + φMAχMA) 1√
2(φMSχMA + φMAχMS)
(8,2) 1 φMSχS φMAχS
(8,2) 1 2 1√2(−φMSχMS + φMAχMA) 1√
2(φMSχMA + φMAχMS)
(8,2) 1 φMAχA φMSχA
(1,4) 1 φAχMA φAχMS
(1,2) 1 2 φAχMA φAχMS
A (10,2) 1 φSχA
(8,4) 1 1√2(φMSχMA − φMAχMS)
(8,2) 1 2 1√2(φMSχMA − φMAχMS)
(1,4) 1 1 φAχS
(1,2) 1 φAχS
Table 3.4: Local baryon operators classified according to symmetry of the flavor and thefour component Dirac spin. The first element inside the braces in the second columnshows the dimensionality of the SU(3)F representation, while the second element showsthe dimensionality of the Dirac spin projections. The third column is the number ofDirac spin embeddings in non-relativistic (ρ = +) construction, while those constructionsinvolving the lower two components also are shown in the fourth column. The multiplicityof operators in the non-relativistic case is 56S, 70MS, 70MA and 20A, and correspondsto the conventional non-relativistic SU(6) construction. The relativistic construction,which involves both positive and negative parity operators, corresponds to the reductionof SU(12). Table taken from [101].
3.4.5 Spatial projection operator symmetries
In order to project the higher spin baryons (J > 3/2) and the excited baryon states, one
needs to employ baryon operators that create states with angular and radial excitations.
Such excitations can be explored using baryon operators with non-local behavior. Covari-
ant derivatives defined in Ref. [100] are incorporated into the three quark operators in
order to obtain suitable representations that transform like orbital angular momentum.
First the covariant derivatives are combined in the definite symmetries with respect to
Excited state spectroscopy 42
their action on the three quark fields. At the single derivative level, there are two relevant
symmetry combinations transforming like an L = 1 object. They are
L = 1; D[1]MS(m) =
1√62D(3)
m −D(1)m −D(2)
m ,
D[1]MA(m) =
1√2D(1)
m −D(2)m , (3.10)
where the different m-components in D(q)m , which acts on the q-th quark, are given by
D(q)±1 = ± i
2(D(q)
x ± iD(q)y ), D
(q)0 = − i
2D(q)z (3.11)
The superscript inside the square brackets in eq. (3.10), stands for the number of deriva-
tives in the operator being represented. There is no totally antisymmetric construction
of the one derivative and the symmetric combination D(1) +D(2) +D(3) is a total deriva-
tive that gives zero when applied to a baryon with zero momentum. Totally symmetric
baryon operators with one derivative are constructed by applying eq. (3.10) to the mixed
symmetric spin-flavor operators as given in Table 3.4.
ψ[1]S =
1√2
(D
[1]MSψ
[0]MS +D
[1]MAψ
[0]MA
), (3.12)
where the superscripts in brackets indicate the number of derivatives in the operator.
The two derivative baryon operators are constructed as follows. Initially, definite
symmetry combinations of two derivatives are formed using the single derivatives, based
on the same general rules of combination of two symmetry labels (see eq. (3.8)). The
three quark symmetry combinations with two derivatives that transform like L = 0, 1, 2
are as follows,
L = 0, 2;D[2]S =
1√2D[1]
MSD[1]MS +D
[1]MAD
[1]MA,
L = 0, 2;D[2]MS =
1√2−D[1]
MSD[1]MS +D
[1]MAD
[1]MA,
L = 0, 2;D[2]MA =
1√2D[1]
MSD[1]MA +D
[1]MAD
[1]MS,
L = 1;D[2]A =
1√2−D[1]
MSD[1]MA +D
[1]MAD
[1]MS. (3.13)
Excited state spectroscopy 43
Using these two derivative symmetric constructions one can construct the totally
symmetric baryon interpolators by applying them on the local baryon operators. They
are as follows.
ψ[2]S = D
[2]S ψ
[0]S ;
1√2
(D[2]MSψ
[0]MS +D
[2]MAψ
[0]MA); D
[2]A ψ
[0]A , (3.14)
where no total derivatives are considered in these constructions. In Table 3.5, we give
the different derivative operators that can be constructed using two derivatives.
L=0,2 L=1S (qqq)1c 0M (qqq)1c [(qqq)8c G8c ]1cA 0 (qqq)1c
Table 3.5: Summary of the bound state interpretations of the two derivative structures.Table taken from Ref. [108]
The subset of operators, as given in Table 3.5, that follows the MS and MA con-
structions, with angular momentum, L = 1, corresponds to a commutation of the two
gauge-covariant derivatives acting on the same quark field [108]. Note that such opera-
tors form color singlet object, [(qqq)8cG8c ]1c , as a combination of the three quarks in color
octet and a gluon field, G. Such operators vanish in a theory without the gauge fields
and corresponds to the chromomagnetic components of the gluonic field strength tensor.
This leads to the identification of the so-called hybrid operators, as they are formed out
of a hybrid combination of quark and gluon fields. As will be observed later, operators
that are proportional to the gluon field strength tensor are essential in obtaining some
states in the spectrum.
These spatial projection operators with definite symmetries are then combined with
the symmetry combinations constructed out of the Dirac spin projection operators using
the standard Clebsch-Gordan formula of SU(2) in order to obtain operators with good J
in the continuum. Table 3.6 shows the details of the allowed spin-parity patterns of the
non-relativistic local and non-local operators for various symmetries in the flavor sector.
We show the constructions up to two derivatives. Using non-relativistic components
alone, it is not possible to construct a negative parity state with a spin higher than 5/2,
even with operators that include two derivatives. Use of relativistic operators along with
non-relativistic ones enables us to extract states with spin up to 7/2 for both the parities.
We follow the same naming convention as used in Ref. [101]. As an example, Ωcss oper-
Excited state spectroscopy 44
ator with spin and parity of the three quarks as 32
−, with two derivatives coupled into L =
2 and total spin and parity JP = 72
−, is denoted as
(Ωcss,S ⊗ (3
2
−1,M
)⊗D[2]L=2,M
)JP= 72
−
,
where the subscripts show the definite symmetries in different labels. Accounting for
the orbital motion classifies the total symmetry in a non-relativistic approach to be
SU(6) ⊗ O(3) super-multiplets with the O(3) describing the orbital motion. With four
component Dirac spinors, the construction of the operators corresponds to SU(12)⊗O(3).
3.4.6 Subduction into lattice irreps
Discretization reduces the symmetry of the space from O(3) to ODh . The eigenstates of
the lattice Hamiltonian transforms as irreps of the ODh . With the reduced symmetry,
the continuum states get separated across the lattice irreps. Restricting the operators in
the continuum to the symmetries allowed by the lattice generates the representations of
these continuum operators that are reducible under ODh . Subduction is this method of
generating the representations of the continuum irreducible operators, which are reducible
under ODh in terms of the lattice irreps. Once the continuum operators are constructed,
we subduce them to the irreps of the octahedral group.
The octahedral group consists of 24 group elements, each corresponding to a rotation
that leaves invariant a cube, or an octahedran embedded within the cube. When the
objects that are rotated involve half-integer values of angular momentum, the number of
group elements doubles to extend the range of rotational angles from 2π to 4π, forming
doubled-valued representations of the octahedral group, giving the irreps for half integer
spins also. Spatial inversion commutes with all rotations and together with identity forms
a two-element point group simply doubling the number of group elements.
The eight irreps of the double-valued representations of the octahedral group include
A1, A2, E, T1 and T2 for integer spins and G1, G2 and H for half-integer spins. The
continuum spins are distributed over these eight irreps on a lattice. Since we consider
only baryons, we discuss only the appearance of the continuum spins over different lattice
irreps for half integer spins (G1, G2 and H). The irreps of the octahedral group are
determined only by the s-spins and the derivative structures. A J = 1/2 spin state
appears only in G1 irrep. A spin of J = 3/2 appears only in H irrep. A continuum spin
of J = 5/2 distributes itself across G2 and H irrep, while J = 7/2 appears over G1, G2
Excited state spectroscopy 45
N(JP )
F ND L S 12
+ 32
+ 52
+ 72
+ 12
− 32
− 52
−
SF 0 0 32
1
1 1 12
1 1
2 0 12
1
2 0 32
1
2 1 12
1 1
2 2 12
1 1
2 2 32
1 1 1 1
AF 1 1 12
1 1
2 0 12
1
2 1 32
1 1 1
2 1 12
1 1
2 2 12
1 1
MF 0 0 12
1
1 1 12
1 1
1 1 32
1 1 1
2 0 12
1
2 0 12
1
2 0 32
1
2 1 12
1 1
2 1 32
1 1 1
2 1 12
1 1
2 2 12
1 1
2 2 32
1 1 1 1
2 2 12
1 1
Table 3.6: The number of operators of a given JP that can be constructed from up totwo derivatives acting on non-relativistic quark spinors. ND indicates the number ofcovariant derivatives, S indicates the total spin of the quarks and L indicates the totalorbital angular momentum. The row indicated in bold face contains the two-derivativehybrid operators, which vanish in the absence of a non-trivial gluon field.
and H. A simple mathematical notation of this phenomena of subduction is given by
ΩJnΛ,r =
∑m
SJ,mnΛ,r ΩJm, (3.15)
Excited state spectroscopy 46
where ΩJnΛ,r is the lattice operator that is the rth row of the nth embedding of the irrep:
Λ on the continuum spin J , subduced from a continuum spin operator ΩJm. The SJ,mnΛ,r
are called the subduction coefficients.
Ωccc G1 H G2
g u g u g uTotal 20 20 33 33 12 12
Hybrid 4 4 5 5 1 1NR 4 1 8 1 3 0
(a)
Ξcc, Ωcc G1 H G2
(Sf +MF ) g u g u g uTotal 55 55 90 90 35 35
Hybrid 12 12 16 16 4 4NR 11 3 19 4 8 1
(b)
Table 3.7: The number of lattice operators for triply (left) and doubly (right) charmbaryons obtained after subduction to various irreps of operators with up to two covariantderivatives. The number of non-relativistic (NR) and hybrid operators for each irrepsand for both parities are given.
Σc, Ωc G1 H G2
(Sf +MF ) g u g u g uTotal 55 55 90 90 35 35
Hybrid 12 12 16 16 4 4NR 11 3 19 4 8 1
(a)
Λc G1 H G2
(MF + AF ) g u g u g uTotal 53 53 86 86 33 33
Hybrid 12 12 16 16 4 4NR 10 3 17 4 7 1
(b)
Ξc G1 H G2
(SF +MF + AF ) g u g u g uTotal 116 116 180 180 68 68
Hybrid 24 24 32 32 8 8NR 23 6 37 10 15 2
(c)
Table 3.8: The number of lattice operators for different singly charm baryons obtainedafter subduction to various irreps of operators with up to two covariant derivatives. Thenumber of non-relativistic (NR) and hybrid operators for each irreps and for both paritiesare given.
Considering the ρ-parity involved, one can classify these irreps into even and odd
parity versions, indicated by the subscripts g (gerade) and u (ungerade). The parity of
the baryon operator is determined from it ρ × (−1)L, where ρ = ρ1ρ2ρ3 and L is the
Excited state spectroscopy 47
angular momentum from the derivatives and ρi is the ρ-value of the ith quark. For non-
relativistic operators, ρ1 = ρ2 = ρ3 = 1 always, hence its parity is solely determined by
the orbital angular angular momentum.
Table 3.7 and Table 3.8 shows the total number of relativistic and non-relativistic oper-
ators constructed using the above procedure for various charm baryons. They also contain
the number of non-relativistic and hybrid operators used for various charm baryons.
3.4.7 Distillation
Having constructed a large set of operators, it is possible to calculate their diagonal and
off-diagonal correlation functions to extract excited states by using variational methods.
However, construction of correlation functions with so many operators, particularly for
the operators with one or more derivatives, is very computationally intensive as it needs
extra inversions for non-local sources. The total time needed for the construction of
correlation function could be as intensive as gauge field generations. To avoid this we
use a recently developed novel technique, called the “distillation method” [96]. Using
this method, one can efficiently construct a large number of correlation functions with
multiple operators. Moreover, this method automatically includes a smearing process, a
requirement for suppressing the high frequency modes that do not contribute significantly
to the low energy regime in the spectrum and thus increases the relative contributions
from the low lying states in the correlation functions. As a smearing function, the con-
straint on the distillation operator is that it should preserve as many symmetries as
possible while effectively removing the contribution from the short distance modes.
A gauge covariant source, with a shape similar to a Gaussian is obtained by Jacobi
smearing [156], where one uses a gauge covariant second order three dimensional lattice
Laplacian operator,
−∇2xy(t) = 6δxy −
3∑j=1
Uj(x, t)δx+j,y + U †j (x− j, t)δx−j,y, (3.16)
where the gauge fields may be constructed from an appropriate covariant gauge field
smearing algorithm. The high energy modes of this operator is suppressed by exponenti-
ating it, exp (σ∇2(t)), with a smearing weight, σ. The resulting smoothened operator is
applied on the quark fields (ψ) to construct the smeared quark fields (ψ). The suppres-
sion of the high energy modes of the Jacobi smearing operator means that only a small
number of modes contribute significantly to the smeared quark fields, ψ.
Excited state spectroscopy 48
The distillation technique [96] replaces the exponentiation of the Laplacian by an outer
product over the low lying eigenmodes of the discretized gauge covariant Laplacian,
xy(t) = Vxz(t)V†zy(t) =
N∑k=1
ξ(k)x (t)ξ(k)†
x (t), (3.17)
where ξ(k)x are a finite number, N << N3
sNc, of eigenvectors of ∇2 evaluated on the
background of spatial gauge fields of the time slice t, once the eigenvectors have been
sorted by the eigenvalues. The number of modes, N (volume dependent), should be
sufficient to sample the required angular structure of the low lying hadronic states, but is
small compared to the number of sites on a time slice. Once the eigenvectors are sorted
by the eigenvalues, in terms of V (t), the kth column of V (t) contains the kth eigenvector
of ∇2 evaluated on the background of the spatial gauge fields of time-slice t. This is the
projection operator onto the subspace, VN , spanned by the N lowest lying eigenmodes,
so 2 = . When N is the same as Nc times the number of sites on a time-slice, then the
distillation operator becomes identity, and the fields acted upon are unsmeared. In this
calculation, we used 64 eigenvectors from four time slices in construction of the baryon
correlation functions.
The Laplace operator inherits many symmetries of the vacuum. It transforms like a
scalar under rotations, is covariant under gauge transformation and is parity and charge
conjugation invariant. If the action of one of these symmetries on the Laplace operator
maps ∇2 onto ∇2, then there is a unitary transformation, R on the vector space spanned
by the eigenmodes of the Laplacian such that
R∇2R† = ∇2. (3.18)
This implies that if v is an eigenvector of ∇2, then Rv is an eigenvector of ∇2.
Considering the definition of the distillation operator, the transformed operator must
obey
RR† = , (3.19)
and so the correlation functions constructed using the distilled fields have same symmetry
properties on the lattice as those constructed using the Laplacian methods.
Excited state spectroscopy 49
Distilled baryon correlation functions
Here we discuss the application of the distillation technique in the construction of the
hadron correlation functions and a smart representation of the correlation functions in
terms of the eigenvectors of the discretized gauge covariant Laplacian, which will make
their computation efficient. We take the case of the isospin-12
baryon as an illustrative
example. A generalization is straight forward. A general baryon annihilation operator
can be written as
Oi(t) = εabcSiαβγ(Γ1d)aα(Γ2u)bβ(Γ3u)cγ, (3.20)
where u and d are the quark fields; Γj is a spatial operator, including possible spatial
displacements, acting on quark j; a, b, c are the color indices, and α, β and γ are the
spin indices; S encodes the spin structure of the operator, and is constructed so that the
operator has the desired quantum numbers. Using eq. (3.20), we construct the baryon
correlation function as
Cij(t′, t) = εabcεabcSiαβγS
∗jαβγ〈[(Γ1d)aα(Γ2u)bβ(Γ3u)cγ(t
′)].[(dΓ1)aα(uΓ1)aα(uΓ1)aα(t)]〉,(3.21)
where the bar over S and Γ indicate these are from the creation operator. Inserting the
outer product representation for distillation operator, we can write the baryon correlator
as
Cij(t′, t) = Φ
i,(p,q,r)αβγ (t′)Φ
j,(p,q,r)
αβγ(t)× [τ ppαα(t′, t)τ qq
ββ(t′, t)τ rrγγ(t
′, t)− τ ppαα(t′, t)τ qrβγ(t′, t)τ rq
γβ(t′, t)],
(3.22)
where Φi,(p,q,r)αβγ = εabcSiαβγ(Γ1ξ
(p))a(Γ2ξ(q))b(Γ3ξ
(r))c (3.23)
and τ ppαβ
(t′, t) = ξ†(p)(t′)M−1αβ (t′, t)ξ(p). Φ
i,(p,q,r)αβγ has a well defined momentum and all
the information about the quantum numbers of the operator goes into this quantity.
While τ ppαβ
(t′, t) is called the operator independent ‘perambulator’ with no explicit mo-
mentum projections. Hence the perambulators can be computed independently between
any source and sink. In principle, once the τ has been computed and stored, the corre-
lation of any local or non-local source and sink operators can be computed.
Note that both the Φ’s and τ ’s are 4N × 4N matrices. Such a decomposition of
the correlation functions reduces the cost of their computation substantially. Firstly,
such a decomposition enables efficient computation of correlation functions with a large
basis of interpolating operators at both the source and the sink time slices. It also en-
ables a momentum projection at both the source and the sink time slices, in contrast
Excited state spectroscopy 50
with conventional spectroscopy, where a momentum projection only at sink is possible.
Furthermore, it also reduces the costs of computations involving all-to-all quark propa-
gators, since calculation of all elements of a perambulator requires only 4N inversions of
the Dirac matrix, in contrast to 12N3s inversions for conventional all-to-all quark propa-
gators. Thus, the ‘distillation’ technique allows one to perform computations involving
disconnected diagrams, multihadron operators, etc. also, with limited computation re-
sources.
3.5 Variational fitting technique
After constructing two point correlation functions (see eq. (2.31)) for a large basis of
interpolating operators, the next task is the extraction of the spectrum from these matrix
of correlation functions in a reliable way. We employ a variational fitting method [97, 98,
99] for this purpose. The construction of the large basis of operators has been discussed
in detail in the previous section. The basic idea of variational fitting technique is to solve
a generalized eigenvalue problem of the form
C(t)vα = λαC(t0)vα, (3.24)
where C(t) is the matrix of correlators at time-slice t. The orthonormality condition
for the generalized eigenvectors is v†αC(t0)vβ = δαβ. The ‘principal correlators’ λα, as
a function of the times-lice, representing the physical two point correlation functions,
are formed from the generalized eigenvalues of the correlation matrices on the respective
time-slices and can be shown to behave at large times as
λα(t) = e−mα(t−t0)(1 +O(e−|δm|(t−t0))), (3.25)
where mα is the mass of a state labeled by α and δm is the energy splitting from the
nearest excited state to α. The eigenvectors form a dim(C) × dim(C) matrix V that
transform the interpolating operators into an optimum linear combination to overlap
with the dim(C) lightest states accessible to them. These eigenvectors should be time
independent to the extent that the correlator is saturated by the lightest dim(C) states.
In practice, one solves the eigenvalue problem on each time-slice and obtain eigenvectors
that can vary with t. The C(t0)-orthogonality of these eigenvectors are then utilized to
order these eigenvectors across the time slices. Examining the form of the eigenvalue in
Excited state spectroscopy 51
eq. (3.24) with the substitution of the spectral decomposition in eq. (2.32) one finds that
Zαi = (V −1)αi
√2mαe
mαt0/2. The inverse of this eigenvector matrix is trivial to compute
owing to the orthonormality property V †C(t0)V = I ⇒ V −1 = V †C(t0).
0.8
0.95
1.1
1.25
1.4
1.55
1.7
1.85
2
Hg (State 0)
χ2/Ndof = 22/15
m= 0.8345(11)
exp(m
n(t
-t0))λ
n(t
)
Hg (State 1)
χ2/Ndof = 9.3/6
m= 0.9315(55)
Hg (State 2)
χ2/Ndof = 27/16
m= 0.9513(22)
0.8
0.95
1.1
1.25
1.4
1.55
1.7
1.85
2
0 5 10 15 20 25
Hg (State 4)
χ2/Ndof = 30/16
m= 0.9575(23)
exp(m
n(t
-t0))λ
n(t
)
t/at
0 5 10 15 20 25
Hg (State 7)
χ2/Ndof = 5.9/12
m= 1.0678(69)
t/at
0 5 10 15 20 25
Hg (State 8)
χ2/Ndof = 11/9
m= 1.1636(76)
t/at
Figure 3.3: Principal correlator fits for six Ωccc states in irrep Hg that will be identified asJP = 3/2+ based on the procedure described in the Section 3.7. Fits are obtained usingeq. (3.27). Data points are obtained from emn(t−t0)λn(t) and the lines show the fits and onesigma-deviation according to the fitting form emn(t−t0)λn(t) = 1−An+Ane
−(m′n−mn)(t−t0),with t0 = 10; the gray points are not included in the fits.
The selection of the parameter t0 plays a crucial role as in practice one works with a
limited number of interpolating operators. The eigenvectors are forced by the solution
procedure to be orthogonal on the metric C(t0), which will be a good approximation to the
true orthonormality, defined in continuum with an infinite number of states and operators,
if the correlator at t0 is dominated by the lightest dim(C) states. The choice of the
optimum t0 is governed by two factors. The contribution from the excited states pushes
the optimum value of t0 to larger values where they would have died down significantly
in the correlators, while the fall in the signal-to-noise ratio with the Euclidean time slice
Excited state spectroscopy 52
urges one to keep t0 to be as small as possible. A remedy for this is to define a χ2
like quantity, so as to gauge how well the generalized eigenvalue problem describes the
correlators for any given t0, as [99]
χ2 =1
12N(N + 1)(tmax − t0)− 1
2N(N + 3)
∑i,j≥i
tmax∑t,t′=t0+1
(Cij(t)−Crec.ij (t))C−1ij (t, t′)(Cij(t
′)−Crec.ij (t′))
(3.26)
where N = dim(C) and C is the data covariance matrix for the correlator Cij com-
puted with jackknife statistics. For a given value of t0, one gets the estimates for the
masses and the overlap factors (Z) by solving the eigenvalue problem and then fitting
the principal correlators. Using these mass and overlap factor estimates one can recon-
struct the correlator matrix using the spectral decomposition in eq. (2.31). Crec.ij are the
correlation functions obtained from the reconstruction procedure discussed above. The
optimum value of t0 is chosen based on the minimum of the chi-square like quantity
defined in eq. (3.26).
After the diagonalization procedure to obtain the optimum linear combination repre-
senting the physical states, we extract the energy of a state by fitting the dependence of
λn on t− t0 to the form,
λn(t, t0) = (1− An)e−mn(t−t0) + Ane−m′n(t−t0), (3.27)
with three fit parameters mn,m′n and An. We find that allowing a second exponential
stabilizes the fits and the resulting second exponential decreases rapidly with large t0,
implying that m′n >> mn. In Figure 3.3, we plot some examples of fits to six Ωccc
principal correlators in irrep Hg, where the fitted states will be identified with JP = 32
+.
The fits approach the constant value, 1 − An, for large t, and they approach one if a
single exponential dominates. This is seen to be the case for most of our fits.
3.6 Rotational symmetry and cross correlations
After the construction of the matrix of two point correlation functions for a large basis
of interpolating operators and the extraction of the physical states employing a reliable
advanced fitting technique, what remains is to make a precise determination of the quan-
tum numbers of the extracted physical states. One primary goal of lattice calculations
such as this one is such an identification of the continuum quantum numbers for the
extracted physical states. As discussed earlier, lattice possess only a reduced symmetry
Excited state spectroscopy 53
nr-nh
nr-h
r-nh r-h
nr-nh
r-nh
r-nh
nr-nh
r-nh
nr-nh
nr-h
r-nh
r-h
nr-nh
r-nh
r-nh
nr-nh
r-nh
32 52 72
32
52
72
t=
5
0.17
0.34
0.50
0.67
0.83
1.00
Figure 3.4: The normalized correlation matrix, Cij/√CiiCjj, of ccc baryon at t/at = 5
for the Hg irrep averaged over all the configurations. Operators are ordered such thatthose subduced from spin 3/2 appear first followed by spin 5/2 and then spin 7/2.
of the continuum. And hence the physical states in the continuum gets subduced over
the lattice irreps. In the continuum limit, the rotational symmetry should be restored.
If the lattice is fine enough, one should be able to observe the effects of the restoration
of the rotational symmetry. The use of smeared fields in construction of interpolating
operators should help in making this restoration more directly observable. With the aim
of creating a more direct link between the lattice operators and the physical quantum
numbers, operators were constructed first in the continuum and then subduced onto the
lattice irreps. Therefore, it is useful to determine whether these lattice operators exhibit
a remnant of the continuum rotational symmetry on the lattice.
To check this, in Figure 3.4, we plot the normalized two point correlation functions
(Cij/√CiiCjj) of triply charm baryons in the Hg irrep at time separation 5 averaged
Excited state spectroscopy 54
n1n2
r1 r2 n1
r1 r2 n1r1n1
n2
r1 r2 n1n2
r1 r2 n1r1
n1n2
r1
r2
n1
r1
r2
n1r1
n1
n2
r1
r2
n1n2
r1
r2
n1r1
32H10FL 52H10FL 72H10FL
32H10FL
52H10FL
72H10FL
32H8FL 52H8FL 72H8
FL
32H10FL
52H10FL
72H10FL
Figure 3.5: The normalized correlation matrix, Cij/√CiiCjj, of Ωccs baryon at t/at = 5
for the Hg irrep averaged over all the configurations. Operators are ordered such thatthose subduced from spin 3/2 appear first followed by spin 5/2 and then spin 7/2.
across all the configurations used in the study. The normalization ensures all the diagonal
entries are unity, while cross correlations are always less than 1. There are 33 operators
used in this irrep, including operators up to two derivatives. The solid lines divide these
operators subduced from continuum spins 32, 5
2and 7
2, and the dashed lines are used for
separating operators defined above with various abbreviations (nr : non-relativistic, nh :
Excited state spectroscopy 55
n1
n2
r1 r2 n1
r1 r2 n1r1n1
n2
r1 r2 n1
n2
r1 r2 n1
r1
n1
n2
r1
r2
n1
r1
r2
n1r1
n1
n2
r1
r2
n1
n2
r1
r2
n1
r1
32H10FL 52H10FL 72H10FL
32H10FL
52H10FL
72H10FL
32H8FL 52H8FL 72H8
FL
32H10FL
52H10FL
72H10FL
Figure 3.6: The normalized correlation matrix, Cij/√CiiCjj, of Ωcss baryon at t/at = 5
for the Hg irrep averaged over all the configurations. Operators are ordered such thatthose subduced from spin 3/2 appear first followed by spin 5/2 and then spin 7/2.
non-hybrid, r : relativistic and h : hybrid). A breaking of the rotational symmetry on the
lattice is expected to reflect in these plots in terms of the relatively large magnitude of the
cross correlations between lattice operators with different continuum spin in comparison
with the diagonal elements. The matrix plot can be trivially seen to show a block
diagonal pattern in the various continuum spin labels. We observe similar patterns of
block diagonal structure in correlation functions of other irreps as well as correlation
Excited state spectroscopy 56
functions of different irreps in other charm baryons. For example, we show in Figure 3.5
and Figure 3.6 the matrix plot of correlation for the Ωccs and Ωcss baryons. The solid lines
in these plots divides different continuum spin operators with varying flavor symmetry
constructions, while the dashed lines divide various abbreviations (n1 : non-relativistic
and non-hybrid, n2 : non-relativistic and hybrid, r1 : relativistic and non-hybrid, r2 :
relativistic and hybrid). As mentioned above, one can trivially observe a block diagonal
structure along the boundaries of different continuum spins. Similar patterns were also
observed for the light ant the strange baryons [102]. These observations suggest that after
subduction a remarkable degree of rotational symmetry remains in the lattice operators.
And this fact gives us confidence in our ability to make unambiguous spin-identification
for the charm baryons.
This analysis also aids in identifying the mixing between different operators. For
example, it is evident from Figure 3.4 that there is strong mixing between ‘nr-nh’ and ‘r-
nh’ type operators and comparatively weak mixing between ‘h’ and ‘nh’-type operators.
As was observed in Ref. [157], we also found additional suppression of mixing for operators
with a given J , but with different L and S, compared to those with the same J , as well
as the same L and S. For example, for the operators O1 = [(32
+
1,S) ⊗ D[0]
L=0,S]32Hg1 and
O2 = [(32
+
1,S) ⊗ D
[2]L=0,S]
32Hg1 both of which have J = 3/2, L = 0, S = 3/2, the matrix
element M12 = C12/√C11C22 = 0.99. On the other hand, the mixings of these operators
with O3 = [(32
+
1,S) ⊗ D[2]
L=2,S]32Hg1 which has J = 3/2, L = 2, S = 3/2 are M13 = 0.0026
and M23 = 0.0031. This suppression of mixing is however evident only for the non-
relativistic operators. We found that for some relativistic operators mixing enhanced
between operators with same J but different L and S, compared to those with same J ,
L and S. This observation, particularly in the non-relativistic sector can be explained in
terms of the heaviness of the charm/bottom quark. Being heavy, the spin-orbit interaction
is significantly small and hence L and S acts as good quantum numbers separately. Hence
those operators with same L and S values can be expected to show strong correlations as
their quantum numbers match, while a cross correlation between operators with different
L and S may not be significant. Similar observations can also be made in the other
charm baryons also.
These plots also aid in qualitative understanding of how much the SU(4)F symmetry
is violated due to very heavy charm quark mass compared to the light and strange quark
masses. These violations of the flavor symmetry are expected to reflect in these matrix
plots in terms of the cross correlation (mixing) between the operators from different
flavor symmetry combinations. One can easily observe in Figure 3.5 and Figure 3.6 that,
Excited state spectroscopy 57
the non-hybrid operators from the 10F flavor combination, corresponding to S, show
significant mixing with non-hybrid operators in the 8F flavor combination, corresponding
to MS and MA. A similar observation can also be made between the hybrid operators
in two different multiplets. It is also very clear from the figures that this mixing is very
strong in the singly charm baryons. Similar observations are also made in other irreps
and different irreps in other charm baryons.
3.7 Spin identification
One main goal, which is a highly non-trivial task, in these lattice calculations is to ensure
that any state identified can be assigned continuum quantum numbers in a reliable way.
In principle, the spin of a hadron state can be determined by extracting the spectrum at
various lattice spacings and then extrapolating to the continuum limit. In the continuum
limit, where the SO(3) symmetry gets restored, the emergence of energy degeneracies
between the lattice irreps will aid us in identifying the continuum spin of the state.
However, there are difficulties with this procedure. Firstly, this requires high precision
calculations with successively finer lattice spacings, and hence with increasing compu-
tational demand. With the current computational resources, this procedure is hence
impractical. Secondly, in the continuum limit, the spectrum can exhibit degeneracies not
just arising from SO(3) symmetry restoration, but also due to the physical degeneracies,
like the ones due to hyperfine splittings and spin-orbit splittings. This raises the problem
of identifying degeneracies due to lattice discretization unambiguously.
An alternative approach, which was detailed in Ref. [99] and used in the calculations
of light baryons [101, 102], light mesons [103], charm mesons as well as heavy-light mesons
[106, 107], was followed by us in these calculations. The main ingredient is the overlap
factor, Zni , as defined in eq. (2.32). To identify the states with spin 1/2 and 3/2 are
fairly straight forward. A state with spin S should have large overlap factors with those
operators, which are subduced from continuum spin S. The operators with largest overlap
factors to this state not only determine the quantum numbers, but also the spin and
spatial structure of this state. Figure 3.7, Figure 3.8 and Figure 3.9 shows the matrix
plot of normalized Zni values of various states across various operators for Ωccc, Ωccs and
Ωcss baryons respectively. The normalization of Zni is given by
Znimaxn[Zni ]
, so that for a
given operator the largest overlap across all states is unity. It is very easy from this
plot to identify the spin of the state from the operators corresponding to darker pixels
for a given state. As one can see, that the low lying states correspond to first seven
Excited state spectroscopy 58
nr-nh
nr-h
r-nh r-h
nr-nh
r-nh r-h
nr-nh
r-nh32 52 72
0
1
2
3
4
5
6
7
8
9
10
11
<----- Operators ----->
<-----States
Figure 3.7: “Matrix” plot of values of overlap factor, Zni , of an operator i to a given state
n, as defined by eq. (2.32) for Ωccc baryon. Zni are normalized according to
Znimaxn[Zni ]
, so
that for a given operator the largest overlap across all states is unity. The normalizedmagnitudes of various operator overlaps to states in the Hg irrep are shown. Darker pixelindicates larger values of the operator overlaps. Various type of operators, for example,non-relativistic (nr) and relativistic (r) operators, as well as hybrid (h) and non-hybrid(nh) operators are indicated by column labels. In addition, the continuum spins of theoperators are shown by 3/2, 5/2 and 7/2. Here one can associate state 6 with quantum
number JP = 72
+, the states 3, 5 and 10 with JP = 5
2
+, and the rest with JP = 3
2
+. The
states 7 and 9 are predominantly hybrid in nature, while states 4, 8 and 10 are found tohave substantial overlap with non-hybrid operators.
states in Ωccc have very strong non-relativistic content. They also have contribution
from the relativistic operators. The higher lying states can be seen to have very little
overlap with the non-relativistic operators, while they overlap very significantly with the
relativistic operators. Similar observation can also be made to identify the low lying
states as strongly non-relativistic in nature for the other charm baryons, which is evident
from the plots for Ωccs and Ωcss baryons. For states with spin 1/2 and 3/2, which gets
subduced simply over single irrep, G1 and H respectively, this procedure is sufficient for
identification.
One can also identify certain states in these plots that have overlap with 5/2 and
7/2 spin, and not with the 3/2. However, spin - 5/2 and 7/2 states appear in multiple
Excited state spectroscopy 59
n1
n2 r1 r2 n1 r1 r2 n1 r1n1
n2 r1 r1 n1
n2 r1 r2 n1 r1
32H10FL 52H10FL 72H10FL
32H8FL 52H8FL 72H8
FL
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
<----- Operators ----->
<-----States
Figure 3.8: Same as in Figure 3.7, but for Ωccs baryon.
lattice irreps, as in these cases, the continuum operator is subduced to multiple lattice
irreps. Identification of spin of these states is made by identifying the Zni degeneracies
n1n2 r1 r2 n1 r1 r2 n1 r1n1
n2 r1 r1 n1n2 r1 r2 n1 r1
32H10FL 52H10FL 72H10FL
32H8FL 52H8FL 72H8
FL
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
<----- Operators ----->
<-----States
Figure 3.9: Same as in Figure 3.7, but for Ωcss baryon.
Excited state spectroscopy 60
0
0.025
0.05
0.075
0.1
0.125
0.15
0.175
0.2
5/2+, state 1
(1/2 +)1,M ⊗
D [2]
L=2,M
100 (1/2 +)2,M ⊗
D [2]
L=2,M
100 (1/2 +)1,S ⊗D [2]
L=2,S
100 (1/2 +)3,S ⊗D [2]
L=2,M
10 (3/2 +)1,M ⊗
D [2]
L=2,M
(3/2 +)1,S ⊗D [2]
L=2,S
1000 (3/2 +)2,S ⊗D [2]
L=2,S
(3/2 -)1,M ⊗
D [1]
L=1,M
Hg: 0.949(3)
G2g: 0.952(2)
0
0.025
0.05
0.075
0.1
0.125
0.15
0.175
0.2
7/2+
100 (3/2 +)1,M ⊗
D [2]
L=2,M
(3/2 +)1,S ⊗
D [2]
L=2,S
100 (3/2 +)2,S ⊗
D [2]
L=2,S
G1g: 0.951(2)
Hg: 0.954(2)
G2g: 0.952(3)
Figure 3.10: A selection of Z-values for Ωccc baryon states conjectured to be identifiedwith JP = 5
2
+(left) and 7
2
+(right). Operators in consideration, which overlap to these
states, are mentioned as the x-axis tick labels. Z-values obtained for a given operator,but from different irreps, are found to be consistent to each other which helps to identifythe spin of a given state.
between states across different irreps. In the continuum limit, for a given physical state,
the overlap factors obtained from different operators, which are subduced from the same
continuum operators should become equal. They should nearly be equal even on a finite
lattice, if the rotational symmetry breaking effects are small. Hence in order to confirm
the identification of a state with a given spin > 3/2, one has to compare the magnitudes
of overlap factors of those operators, which are subduced into different irreps. In Figure
3.10 and Figure 3.11, we show a selection of Z values for a set of Ωccc and Ωccs states
respectively, across a set of operators with matching overlaps. A degeneracy between
the Z-values is clearly evident in these plots helping in identification of the states with
spin > 3/2. After confirming the matching overlap factors, it is also necessary to check
whether the energy of the identified state also matches over the irreps. The legends in
these plots contains the ma values obtained for each of these states in the respective
irreps. Comparing those estimates one can see that the spin identification made in
these plots are valid. In Figure 3.12 and Figure 3.13, we show the histogram plots of
normalized Z values of a selection of low lying Ωccc and Ωccs states identified to be spin
1/2(black), 3/2(red), 5/2(green) and 7/2(blue) for the positive parity, respectively. The
normalization of the Z values follows the same definition as used in the Figure 3.7. Once
this is made, we perform a simultaneous fitting of the respective principal correlators
over the irreps to get the final mass. Figure 3.14 and Figure 3.15 show two examples of
Excited state spectroscopy 61
0
0.01
0.02
0.03
0.04
0.05
10 (10F ) ⊗
(1/2 +)
2,M
⊗ D [2]
L=2,M
10 (10F ) ⊗
(1/2 +)
3,M
⊗ D [2]
L=2,M
10 (10F ) ⊗
(3/2 +)
2,S ⊗ D [2]
L=2,S
(10F ) ⊗
(3/2 -)
1,M
⊗ D [2]
L=1,M
10 (8F ) ⊗
(1/2 +)
1,A ⊗ D [2]
L=2,M
10 (8F ) ⊗
(1/2 +)
3,M
⊗ D [2]
L=2,S
10 (8F ) ⊗
(3/2 +)
1,M
⊗ D [2]
L=1,A
100 (8F ) ⊗
(3/2 +)
1,M
⊗ D [2]
L=1,M
10 (8F ) ⊗
(3/2 +)
1,M
⊗ D [2]
L=2,S
100 (8F ) ⊗
(3/2 +)
1,S ⊗ D [2]
L=1,M
10 (8F ) ⊗
(3/2 +)
2,S ⊗ D [2]
L=2,M
5/2+
Hg: 0.775(3)
G2g: 0.780(2)
0
0.05
0.1
0.15
0.2
0.25
10 (10F ) ⊗
(1/2 -)
1,M
⊗ D [2]
L=2,M
10 (10F ) ⊗
(1/2 +)
3,M
⊗ D [2]
L=2,M
100 (10F ) ⊗
(3/2 -)
1,M
⊗ D [2]
L=1,M
10 (10F ) ⊗
(3/2 -)
1,M
⊗ D [2]
L=2,M
1000 (10F ) ⊗
(3/2 +)
2,S ⊗ D [2]
L=2,S
10 (8F ) ⊗
(1/2 -)
1,S ⊗ D [2]
L=2,M
10 (8F ) ⊗
(3/2 +)
1,S ⊗ D [1]
L=1,M
10 (8F ) ⊗
(3/2 -)
1,M
⊗ D [2]
L=1,A
100 (8F ) ⊗
(3/2 -)
1,M
⊗ D [2]
L=1,M
10 (i8F ) ⊗
(3/2 -)
1,M
⊗ D [2]
L=2,S
100 (8F ) ⊗
(3/2 -)
1,S ⊗ D [2]
L=1,M
1000 (8F ) ⊗
(3/2 -)
2,S ⊗ D [2]
L=1,M
1000 (8F ) ⊗
(3/2 -)
2,S ⊗ D [2]
L=2,M
5/2-
Hu: 0.749(3)
G2u: 0.752(2)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
100 (10F ) ⊗
(3/2 +)
1,M
⊗ D [2]
L=2,M
(10F ) ⊗
(3/2 +)
1,S ⊗ D [2]
L=2,S
10 (10F ) ⊗
(3/2 +)
2,S ⊗ D [2]
L=2,S
100 (8F ) ⊗
(3/2 +)
1,M
⊗ D [2]
L=2,S
(8F ) ⊗
(3/2 +)
1,S ⊗ D [2]
L=2,M
7/2+ G1g: 0.783(3)
Hg: 0.780(3)
G2g: 0.781(3)
0
0.005
0.01
0.015
0.02
(10F ) ⊗
(3/2 -)
1,S ⊗ D [2]
L=2,S
10 (10F ) ⊗
(3/2 -)
2,S ⊗ D [2]
L=2,S
100 (8F ) ⊗
(3/2 +)
2,S ⊗ D [2]
L=2,M
7/2-
G1u: 0.823(5)
Hu: 0.829(5)
G2u: 0.827(5)
Figure 3.11: A selection of Z-values for Ωccs baryon states conjectured to be identifiedwith JP = 5
2
+(top left), JP = 5
2
−(top right), 7
2
+(bottom left) and 7
2
−(bottom right).
Operators in consideration, which overlap to these states, are mentioned as the x-axistick labels. Z-values obtained for a given operator, but from different irreps, are foundto be consistent to each other which helps to identify the spin of a given state.
such simultaneous fits performed for a state with spin 5/2 and 7/2 respectively.
3.8 Summary
This chapter discusses various details about the techniques used in the excited charm
baryon spectroscopy calculations. We begin this chapter discussing the conventional
methodology followed in spectroscopy calculations (Section 3.1), following which we dis-
Excited state spectroscopy 62
G1 g
Hg
G2 g
Figure 3.12: Histogram plot showing normalized overlap factor, Z, of a few operatorsonto some of the lower-lying Ωccc states in each of the lattice irreps. The overlaps arenormalized according to
Znimaxn[Zni ]
s that the largest overlap across all states for a given
operator is unity. Black bars correspond to spin-1/2 state, red for spin-3/2, green forspin-5/2 and blue represents spin-7/2 states. Lighter and darker shades at the top ofeach bar represent the one-sigma statistical uncertainty.
G1 g
Hg
G2 g
Figure 3.13: Same as the Figure 3.12, but for Ωccs baryon.
cuss about the challenges in the excited state spectroscopy (Section 3.2). In Section 3.3,
3.4 and 3.5 we discuss how the use of anisotropic lattices, the derivative-based operator
construction formalism, and the variational fitting method help us meet these challenges.
Excited state spectroscopy 63
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25
Hg
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25
G2g
Figure 3.14: The joint fits of principal correlators identified to be the same 5/2 stateobtained from Hg and G2g irrep. The red point are those included in the fit, while thegrey points are excluded.
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25
G1g
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25
Hg
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25
G2g
Figure 3.15: The joint fits of principal correlators identified to be the same 7/2 stateobtained from G1g, Hg and G2g irrep. The red point are those included in the fit, whilethe grey points are excluded.
In the last two sections, 3.6 and 3.7, we discuss how a reliable identification of the quan-
tum numbers of a state is made. In short, this chapter discusses all the major technical
details related with the calculations referred in the next chapter.
Chapter 4
Charm baryon spectrum
In this chapter, we present our results on the spectra of charm baryons with spins up
to 7/2 and for both the parities. Results for the extracted masses are quoted in terms
of various energy splittings. This convention is preferable, as it is expected to reduce
the impact of various systematic errors that could arise from the uncertainty in determi-
nation of the quark mass parameters as well as the discretization of the lattice action.
Particularly, we consider the option of subtracting out the valence charm quark content
in the spectrum, as the contribution for the discretization errors from the charm quark
is larger than the light quarks. Various spin dependent energy splittings between the ex-
tracted states are also considered and compared with similar splittings at light, strange
and bottom quark masses, wherever possible. We divide the discussion of results in three
separate sections for the charm baryons with three, two and one flavor in the valence
sector, respectively.
4.1 Triply charm baryons
With three valence charm quarks, triply charm baryons are the heaviest of all the baryons
made of u, d, s and c quarks. The triply-charmed baryons may provide a new window for
understanding the structure of baryons, as pointed out by Bjorken several years ago [25].
A comprehensive study of the excitation spectra of these states, where the complications
of light-quark interaction are absent, can provide information about the quark confine-
ment mechanism as well as elucidating our knowledge about the nature of the strong force
by providing a clean probe of the interplay between perturbative and non-perturbative
QCD [158]. Just as the quark-antiquark interactions are examined in charmonia, these
64
Charm baryon spectrum 65
studies will probe the quark-quark interactions in the charm baryons. The flavor content
being similar to ∆(uuu) and Ω(sss), only symmetric flavor combination is allowed for
the triply charm baryons. Table 3.7(a) shows the total number of operators including up
to two derivatives that we use in our calculations.
4.1.1 The spectrum
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-0.0
0.4
0.8
1.2
1.6
2.0
2.4
m -
3/2
mη
c
[G
eV
]
Ωccc
Figure 4.1: Spin identified relativistic spectra of triply-charmed baryons with respectto 3
2mηc . The boxes with thick borders corresponds to the states with strong overlap
with hybrid operators. The states inside the pink ellipses are those with relatively largeoverlap to non-relativistic operators.
Figure 4.1 shows the relativistic spectrum of triply charm baryons as splitting from
3/2 times the mass of ηc meson, while Table 4.1 contains the estimates. The factor of 32
corrects for the different number of valence charm quarks in triply-charmed baryon and
charmonium states. As mentioned earlier, plotting in this splitting is expected to reduce
the possible discretization errors in the estimates. Boxes with thicker borders correspond
to those with a greater overlap onto operators proportional to the field strength tensor
as discussed in the previous chapter and might consequently be hybrid states. The states
inside the pink ellipses have relatively large overlap with non-relativistic operators.
Charm baryon spectrum 66
12
+ 32
+ 52
+ 72
+ 12
− 32
− 52
− 72
−
0.923(13) 0.287(6) 0.930(15) 0.921(49) 0.644(9) 0.648(13) 1.040(64) 1.205(28)0.929(14) 0.841(31) 0.988(15) 1.136(31) 1.186(31) 1.233(25)1.563(33) 0.954(13) 1.155(43) 1.248(44) 1.234(24)1.607(42) 0.989(13) 1.273(21) 1.289(34) 1.236(32)1.992(80) 1.618(40) 1.656(69) 1.823(44)2.236(56) 2.147(46) 1.850(51)
2.165(43)2.298(46)
Table 4.1: Relativistic spectrum mΩccc− 3
2mηc in units of GeV.
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-0.0
0.4
0.8
1.2
1.6
2.0
2.4
m -
3/2
mη
c
[G
eV
]
Figure 4.2: Spin identified non-relativistic spectra of triply-charmed baryons with respectto 3
2mηc . The boxes with thick borders corresponds to the states with strong overlap with
hybrid operators.
An analysis of the spectrum using only the non-relativistic operators is presented in
the Figure 4.2. One can easily see that the low lying states in the relativistic spectrum
(Figure 4.1) looks the same as in the non-relativistic spectrum. This supports the non-
relativistic nature of the low lying spectrum of triply charm baryons. The main difference
between Figure 4.1 and Figure 4.2 is the absence of negative parity states with spin higher
Charm baryon spectrum 67
than 3/2 in the non-relativistic spectrum. This is because with the two component Dirac
spinor (ρ = +1) and using up to two derivatives, one can construct operators that has
continuum quantum numbers only up to spin 3/2 for negative parity. A similar non-
relativistic spectrum was studied for the triply bottom baryons in Ref. [157]. The main
difference with the set of operators used in that calculation is the inclusion of the hybrid
operators in this work, which provides access to two higher lying states with spin 1/2
and 3/2, represented using boxes with thick borders in Figure 4.2.
In the low lying bands of states, up to the second band, the pattern of states expected
based on and SU(6)⊗O(3) symmetry is as mentioned in Table 3.6. With only symmet-
ric flavor structure allowed for triply charm baryons, only a symmetric spin pattern can
combine with a local spatial structure to give a totally symmetric baryon wavefunc-
tion, modulo the color part. Hence the ground state is formed by a spin-3/2 (Hg) local
structure. The first excited band is formed from the states with overlap on to the one
derivative operators. One derivative operators possess only MS and MA construction
with L = 1. Hence they can combine only with MS and MA spin constructions to give
a totally symmetric spin-spatial projection operator. Thus, two negative parity states
(1/2− and 3/2−) are formed in the first excitation band as a result of the spin-orbit
interaction between the spin-1/2 (G1g) structure and the orbital angular momentum of
L = 1. With two derivatives, a set of positive parity states with a range of spins from 12
to 72
is expected in the second excitation band based on a model with SU(6)⊗O(3) sym-
metry. All these patterns up to the second excitation band is clearly evident in the full
spectrum determined by this calculation, as shown in Figure 4.1. This agreement of the
number and pattern of low lying states between the lattice spectra obtained in this work
and the expectations based on non-relativistic quark spins implies a clear signature of
SU(6)×O(3) symmetry in the spectra. Such SU(6)×O(3) symmetric nature of spectra
was also observed in Ref. [102]. It is not meaningful to interpret the higher excited states,
that are extracted in this calculation, in terms of SU(6) × O(3) symmetry due to the
following reasons. We have not included non-relativistic operators with three derivatives
and as the relativistic operators are expected to have strong overlaps with the higher
excited states, it is not clear how the relative importance of all the relevant operators
overlapping to that state will change in the presence of non-relativistic operators having
three or more number of derivatives. For the same reasons, it is also not possible to
identify negative parity states with strong hybrid content.
Charm baryon spectrum 68
4.1.2 Investigating lattice artifacts
For a simulation carried out at a finite lattice cut-off, the results obtained will differ
from their continuum counterparts and the difference can be understood in a Symanzik
expansion in powers of the lattice spacing. Usually, the best means of removing these
artifacts is to perform calculations at a range of lattice spacings and to use the expansion
to extrapolate to the limit of vanishing lattice spacing. In this study, we do not have
sufficient data to carry out this extrapolation. So we follow a calculation similar to the
one that was made in Ref. [106] so as to get an estimate on how much our spectrum is
affected with the corrections from the discretization.
In Ref. [106], a simple experiment was carried out to provide a crude estimate of the
systematic uncertainty due to spatial discretization artifacts. The charm quark action is
discretized using an action that removes the O(as) effects at the tree-level. To assess if
radiative corrections to the co-efficient of the improvement term in the charm-quark action
could lead to significant changes in physical predictions, a second calculation was carried
out after the spatial clover co-efficient was boosted from its tree-level value, cs = 1.35
to cs = 2. In the charmonium study, it was observed that this new ad-hoc value for cs
is found to increase the hyperfine spitting (mJ/ψ −mηc), from its tree level estimate, by
∼ 45 MeV and take it to its physical value. A very similar shift was observed in other low
lying states in the charmonium spectrum indicating a similar magnitude of uncertainty
in their estimation. Higher lying states do not show any statistically significant difference
between the two charm quark actions.
With three valence charm quarks, triply charm baryons is expected to posses the
maximum discretization errors among the charm baryons. We perform a similar calcu-
lation using the charm quark action with cs = 2.0, where we computed the two point
correlation functions for the triply charm baryons. Figure 4.3 and Figure 4.4 shows the
energy splitting, mΩccc − 3/2mηc , of positive and negative parity, respectively, for various
states that we extracted using the two charm quark actions. Here the factor of 32
corrects
for the different number of valence charm quarks in a triply-charmed baryon and the ηc
meson. A very similar shift, to that observed in charmonia, is observed in the lowest
few states, indicating a similar scale of uncertainty in this calculation for triply charm
baryons also. The lowest few states, where this shift is observed, has been shown in
insets of the plots for clarity. The higher lying triply charm states also do not show any
statistically significant shifts between the estimates from the two calculations.
Charm baryon spectrum 69
cs=1.35 cs=2 cs=1.35 cs=2 cs=1.35 cs=2 cs=1.35 cs=20.0
0.4
0.8
1.2
1.6
2.0
2.4
∆m
ass
(G
eV
)
0.28
0.3
0.32
0.34
1/2+
3/2+
5/2+
7/2+
Figure 4.3: Comparison of the positive parity triply charm baryon spectrum from thetwo calculations; one using the tree level tadpole improved spatial clover co-efficient,cs = 1.35, and the other using ad-hoc boosted value, cs = 2.0, to study the systematicscoming from the discretization. The spectrum is in terms of the energy splittings from3/2 times the mass of the ηc meson in the respective calculations.
4.1.3 Valence quark mass dependence of energy splittings
Spin-dependent splittings between the triply-flavored baryons provide an insight into
interactions between three confined quarks with the same mass. We consider now the
dependence of these splittings on the mass of the valence quarks.
As presented in Table 3.6, operators with increasing numbers of gauge-covariant
derivatives (which can correspond to non-zero orbit angular momentum in a quark model)
create states with numerous values of total spin J . For example, we construct flavor (F )
decuplet states with D = 2, S = 32
and L = 2 with the combination 10FS⊗(S)S⊗(D)S,where S in the subscript stands for symmetric combinations, as defined in Ref. [101]. In
this way of construction, we get four quantum numbers with JP = 12
+, 3
2
+, 5
2
+and 7
2
+.
In the heavy quark limit, the spin-orbit interaction vanishes since the interaction term is
inversely proportional to the square of the quark mass. States corresponding to quantum
numbers (|L, S, JP 〉 ≡ |2, 32, 1
2
+〉, |2, 32, 3
2
+〉, |2, 32, 5
2
+〉 and |2, 32, 7
2
+〉) will thus be degener-
ate in the heavy quark limit. Similarly, two states with quantum numbers JP = 12
−and
Charm baryon spectrum 70
cs=1.35 cs=2 cs=1.35 cs=2 cs=1.35 cs=2 cs=1.35 cs=20.0
0.4
0.8
1.2
1.6
2.0
∆m
ass
(G
eV
)
0.6
0.625
0.65
0.675
0.7
0.725
1/2-
3/2-
5/2-
7/2-
(a) (b)(a) (b)
Figure 4.4: Comparison of the negative parity triply charm baryon spectrum from thetwo calculations; one using the tree level tadpole improved spatial clover co-efficient,cs = 1.35, and the other using ad-hoc boosted value, cs = 2.0, to study the systematicscoming from the discretization. The spectrum is in terms of the energy splittings from3/2 times the mass of the ηc meson in the respective calculations.
32
−with L = 1 and S = 1
2will also be degenerate in the heavy quark limit. In Figure 4.5,
we plot absolute values of energy differences between energy levels which originate from
the spin-orbit interaction of the following (L , S) pairs : (2,32
–in the left column), (2,12
–in the middle column) and (1,12
–in the right column). We plot these spin-orbit energy
splittings at various quark masses corresponding to following triple-flavored baryons :
∆uuu, Ωsss, Ωccc and Ωbbb. We identified the states with these (L, S) pairs by finding the
operators, which incorporate these pairs and which have major overlaps to these states.
These energy differences are obtained from the ratio of jackknifed correlators, which in
general, reduce error bars by canceling systematic corrections in these correlators. For
bottom baryons we use data from Ref. [157] and for light and strange baryon results from
Ref. [102] are used.
It should be noticed that energy levels shown for bottom baryons are obtained from
only non-relativistic operators [157]. For charm baryons, we show results from the full
set of operators (including relativistic and non-relativistic operators) as well as from only
Charm baryon spectrum 71
|E(1/2 +) - E(3/2 +
)|
|E(5/2 +) - E(3/2 +
)|
|E(7/2 +) - E(3/2 +
)|
|E(5/2 +) - E(3/2 +
)|
|E(3/2 -) - E(1/2 -
)|
-160.0
-120.0
-80.0
-40.0
0.0
40.0
80.0
120.0
160.0
200.0
240.0
∆m
ass
(MeV
)
Hg,1,S
⊗ D[2]
L=2,SG
1g,1,M⊗ D
[2]
L=2,MG
1g,1,M⊗ D
[1]
L=1,M
Ωccc
(NR + R)
Ωbbb
(NR)
Ωccc
(NR)
Ωsss
∆uuu
Figure 4.5: Energy splittings between states with same L and S values, starting fromlight to heavy baryons. For Ωbbb, results are with only non-relativistic operators [157];for Ωccc, results from relativistic and non-relativistic as well as only non-relativistic op-erators are shown, and for light and strange baryons results are with relativistic andnon-relativistic operators [102]. These results are obtained from fitting the jackknife ra-tio of the correlators which helps to get smaller error bar in splittings. The left columnis for the states with D = 2, S = 3
2and L = 2. The symbol Hg,1,S refers to the first em-
bedding of irrep Hg in the totally symmetric Dirac spin combination, while D[2]L=2,S refers
to spatial projection operators with two derivatives in a totally symmetric combination,and with orbital angular momentum two. Similarly, the middle column is for the stateswith D = 2, S = 1
2and L = 2. Here irrep is G1g and both Dirac and derivative are in
a mixed symmetric combination. In the right column these negative parity states haveD = 1, S = 1
2and L = 1. Here again irrep is G1g and both Dirac and derivative are in a
mixed symmetric combination.
non-relativistic operators, and for light and strange baryons the splittings results are
obtained also with the full set operators. In some cases, we find that the inclusion of
the relativistic operators increase error bars. As one can observe that the degeneracy
between these spin-orbit split states is more or less satisfied both for bottom and charm
quarks. However, data with higher statistics is necessary to identify the breaking of this
degeneracy at charm quark. For ∆(uuu) and Ω(sss) some of these splittings are non-zero.
This is expected because of the presence of the light quark masses in the denominator of
Charm baryon spectrum 72
the spin-orbit interaction, which enhance these splittings.
700
800
900
1000
1100
1200
1300
1400
0.1 1 10 100
∆m
(M
eV
)
mps2 (GeV
2)
1/2+
0
200
400
600
800
1000
1200
1400
1600
0.1 1 10 100
∆m
(M
eV
)
mps2 (GeV
2)
3/2+
700
800
900
1000
1100
1200
1300
1400
0.1 1 10 100
∆m
(M
eV
)
mps2 (GeV
2)
5/2+
700
800
900
1000
1100
1200
1300
1400
0.1 1 10 100
∆m
(M
eV
)
mps2 (GeV
2)
7/2+
Figure 4.6: Energy splittings of a few positive parity triple-flavored baryons from vectormeson (T−−1 ) ground state are plotted against the square of the pseudoscalar masses.Data at charm quark are from this work, while light and strange quark results are fromRef. [102] which uses the same gauge field configurations. Data points for bottom quarkare shown by open symbols and are taken from Ref. [157], which uses non-relativisticaction. A factor 3/2 is multiplied with vector meson masses to account for the differencein the flavor content between baryons and mesons. The fits are performed assuming aform a+ b/mps, inspired by HQET [39], (except for 3
2
+ground state, where data are also
fitted with a constant, which gave better χ2/dof). The light quark data has not beenconsidered in the fits though the fit estimates are extrapolated to light quark masses.
We also compared how energy splittings change between light and heavy baryons.
Some of these, such as the hyperfine splitting, vanish in the heavy quark limit while others
become constant. However, most splittings tend to be higher at lighter quark masses
where relativistic effects are prominent. We determined the energy difference between
the triply-flavored baryons with respect to the vector mesons with two constituents of
the same flavor. To make a comparison, which is independent of the quark mass in the
heavy quark limit, we subtract 32
of the vector meson mass, where this factor simply
takes account of the difference in the flavor content between baryons and mesons. It is
Charm baryon spectrum 73
to be noted here that the light and strange vector mesons are isoscalar in nature, while
for the charm and the strange vector mesons the disconnected diagrams were omitted
in the calculations so as to reduce the computational costs. It is expected that the
contributions from such disconnected diagrams will be very small, since such diagrams
are OZI suppressed. Specifically we consider following splittings: m∆uuu− 32mωuu ,mΩsss−
32mφss ,mΩccc − 3
2mJ/ψcc and mΩbbb − 3
2mΥbb
. These splittings mimic the binding energies
of triple-flavored states and thus it is interesting to compare these as a function of quark
masses. In Figure 4.6 and Figure 4.7, we plot these splittings, for the ground-state and
a few excitations as a function of the square of the pseudoscalar meson masses. For
∆++(uuu) and Ωsss baryons, results are from Ref. [102], while for Ωbbb, we use results
from Ref. [157]. Notice that most of the splittings in various spin parity channels decrease
with quark mass.
400
600
800
1000
1200
1400
0.1 1 10 100
∆m
(M
eV
)
mps2 (GeV
2)
1/2−
400
600
800
1000
1200
1400
0.1 1 10 100
∆m
(M
eV
)
mps2 (GeV
2)
3/2−
800
1000
1200
1400
1600
1800
0.1 1 10 100
∆m
(M
eV
)
mps2 (GeV
2)
5/2−
800
1000
1200
1400
1600
1800
0.1 1 10 100
∆m
(M
eV
)
mps2 (GeV
2)
7/2−
Figure 4.7: Energy splittings of a few negative parity triple-flavored baryons from vectormeson (T−−1 ) ground state are plotted against the square of the pseudoscalar masses.Details same as in Figure 4.6. The fits are performed assuming a form a+b/mps, inspired
by HQET [39], except for 12
−ground state, where data are also fitted with a constant,
which gave better χ2/dof .
In the heavy quark limit, naively one can expand the mass of a heavy hadron, with
Charm baryon spectrum 74
State a b χ2/DOF State a b χ2/DOF12
+773 (16) 59 (22) 1.876 1
2
−511 (12) ?????
12
+758 (16) 117 (19) 0.411 1
2
−963 (31) 193 (27) 10.89
32
+168 (9) -2.0 (9) 0.474 3
2
−513 (14) 45 (15) 0.085
32
+170 (5) ?????
32
+640 (22) 179 (21) 0.558 3
2
−1089 (30) 109 (20) 0.200
32
+765 (17) 149 (21) 1.228
32
+805 (12) 177 (21) 0.179
52
+760 (17) 140 (16) 0.009 5
2
−954 (61) 230 (50) 10.33
52
+807 (17) 167 (15) 0.148 5
2
−1021 (26) 271 (26) 28.54
72
+739 (17) 201 (16) 0.687 7
2
−1009 (30) 220 (27) 17.41
Table 4.2: The fit estimates for the parameters in the fit form, a + b/mps (inspired byHQET), for different states from various quantum channels. The states are identifiedbetween this table and the figures 4.6 and 4.7 by keeping the colors consistent betweenthem.
n heavy quarks, as MHnq = n mQ + A + B/mQ + O(1/mQ2) [39]. Though in Ref. [39]
such an expansion is argued for singly charm baryons, here we assume it to work for
hadrons with multiple quarks. Hence, the energy splittings between heavy hadrons, that
take into account the difference in the heavy flavor content, can be expressed in the form
a + b/mQ and similarly in the heavy quark limit with a + b/mps. Note this form is not
expected to be valid for light hadrons. Using this function, we fitted the data obtained
for mΩbbb − 32mΥbb
,mΩccc − 32mJ/ψcc and mΩsss − 3
2mφss . Because of the very different
behavior in the chiral limit, the light quark points, m∆uuu− 32mωuu are excluded from the
fit. For the cases where data for the bottom quark are not available (mainly for negative
parity cases), fitting is done using only the two data at charm and strange masses. While
there is no good reason for the heavy-quark inspired functional form to model the data at
the strange quark mass, a good fit is still found. We also extrapolate fit results to lighter
pion masses and observe that for many cases they pass through the light quark points
though those points are not included in these fits. However, as mentioned before, one
should not use this extrapolation to get splittings at lighter quark masses as is evident in
many cases where data points fall significantly below the fit. It is worth noting that that
the energy splittings for the spin-32
+and spin-1
2
−ground states are almost constant, even
when varying the quark mass from light to bottom. In fact, we observe better fit with a
constant term for these splittings. Table 4.2 tabulates the fit estimates for the parameters
Charm baryon spectrum 75
in the fit form, a+ b/mps (inspired by HQET), for different states from various quantum
channels.
4.2 Doubly charm baryons
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
m -
mη
c
[G
eV
]
Ξcc
Figure 4.8: Spin identified Ξcc baryon spectra for both parities and with spin up to 7/2.Energy splittings of the Ξcc baryons from the mass of mηc meson, which has same numberof charm quarks, are shown here. The states inside the magenta boxes are those withrelatively larger overlap to non-relativistic operators and the states with thick borderscorresponds to the states with strong hybrid content. The number of states inside theseboxes matches with the expectations based on non-relativistic quark spins, as shown inTable 3.6. This agreement of the number of low lying states between the lattice spectraobtained in this work and the expectations based on non-relativistic quark spins impliesa clear signature of SU(6)×O(3) symmetry in the spectra.
Doubly charm baryons are interesting systems as they provide a unique opportunity
to get insights into the nature of strong force in the presence of slow relative motion
of the heavy quarks along with the relativistic motion of a light quark. The study of
their excited state spectra and various energy splittings will help us understand how the
collective degrees of freedom gives rise to excitations in these systems. A comparison of
these excitations with corresponding spectra of singly and triply charm baryons, where the
number of charm quark is one less and more respectively, will be helpful to get information
Charm baryon spectrum 76
about quark-quark interactions within these systems. A doubly-heavy baryon can be
treated as a bound state of a heavy antiquark and a light quark (qQQ ↔ qQ) in the
limit when the typical momentum transfer between the two heavy quarks is larger than
ΛQCD [159, 160]. In this limit of quark-diquark symmetry, definite predictions of the spin
dependent energy splittings can be made [42, 161]. It is thus interesting to study those
spin splittings based on first principles to check whether charm quark is heavy enough
to respect this quark-diquark symmetry.
With two valence charm quarks, doubly charm baryons are the second heaviest
baryons of the SU(4) flavor family. The flavor pattern being similar to the Σ++ baryon
(suu), the allowed flavor symmetry combination are symmetric, mixed-symmetric and
mixed-antisymmetric. Table 3.7(b) shows the total number of operators including up to
two derivatives, we have used in our calculations.
4.2.1 The spectrum
12
+ 32
+ 52
+ 72
+ 12
− 32
− 52
− 72
−
0.676(08) 0.770( 9) 1.329(13) 1.393(10) 1.049(11) 1.098(12) 1.224(14) 1.583(68)1.338(22) 1.291(23) 1.376(14) 1.558(15) 1.083( 8) 1.185(10) 1.572(22)1.342(20) 1.351(33) 1.523(19) 1.201(12) 1.243(13) 1.741(20)1.397(30) 1.391(13) 1.534(16) 1.604(30) 1.653(35) 1.806(34)1.456(21) 1.422(28) 1.602(17) 1.666(32) 1.734(21)1.526(18) 1.446(30) 1.687(44) 1.836(20)1.564(14) 1.541(14) 1.751(33) 1.925(21)1.780(72) 1.555(22) 1.769(28)2.049(37) 1.565(29)2.061(47)2.078(56)
Table 4.3: Relativistic spectrum mΞcc−mηc in units of GeV.
In Figure 4.8 and Figure 4.9, we show the spin identified full spectra, that we extracted
on our lattices, of Ξcc and Ωcc baryons, respectively, as splittings from ηc meson, while
Table 4.4 and Table 4.3 contains the respective estimates. The states inside the magenta
boxes are those with relatively larger overlap to non-relativistic operators and the states
with thick borders corresponds to the states with strong hybrid content.
Similar to the case of the triply charm baryons, an analysis of the number of pattern of
the states in the spectrum for both the doubly charm baryons points to a spectrum based
Charm baryon spectrum 77
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
m -
mη
c
[G
eV
]
Ωcc
Figure 4.9: Spin identified spectra of Ωcc baryons for both parities and with spin upto 7/2. Energy splittings of the Ωcc states from the mass of ηc meson, which has samenumber of charm quark, are shown here. All other details are same as in Figure 4.13.
12
+ 32
+ 52
+ 72
+ 12
− 32
− 52
− 72
−
0.720(7) 0.792(9) 1.362(16) 1.438(12) 1.084(12) 1.134(8) 1.272(11) 1.736(18)1.314(37) 1.345(18) 1.405(14) 1.620(12) 1.088(11) 1.176(23) 1.640(22)1.371(17) 1.364(27) 1.531(21) 1.206(15) 1.222(17) 1.767(24)1.391(35) 1.400(16) 1.569(15) 1.677(27) 1.703(35) 1.786(30)1.462(21) 1.473(35) 1.577(28) 1.736(24) 1.708(33) 1.840(25)1.555(15) 1.475(27) 2.070(48) 1.837(26) 1.709(30)1.563(17) 1.535(22) 1.882(20) 1.920(26)1.896(47) 1.587(17)2.115(49) 1.602(16)2.143(44)2.186(37)
Table 4.4: Relativistic spectrum mΩcc−mηc in units of GeV.
on non-relativistic quark models. With two flavor symmetry constructions possible, dou-
bly charm baryons allows two local non-relativistic operators : one from the combination
of spin 1/2 (G1g) and the other from the combination of spin 3/2 (Hg) with local spa-
tial structure, giving rise to two positive parity operators with total angular momentum
1/2 and 3/2, respectively. While the set of one derivative operators posses a pattern of
Charm baryon spectrum 78
(3,3,1,0) with the total angular momentum J = (12, 3
2, 5
2, 7
2), the two derivative operators
form a pattern of (6,8,5,2) with the total angular momentum J = (12, 3
2, 5
2, 7
2). Figure
4.8 and Figure 4.9 shows this pattern is very clearly followed in the low lying three bands
with the allowed quantum numbers as mentioned above. It is to be pointed out that in
our calculation, we use the whole set including non-relativistic and relativistic operators
and still we obtain the number and pattern of states allowed by purely non-relativistic
operators. This agreement of the number of low lying states between the lattice spectra
obtained in this work and the expectations based on non-relativistic quark spins implies
a clear signature of SU(6)×O(3) symmetry in the spectra. As mentioned in the previous
section, such a symmetry based on non-relativistic quarks is observed in the triply charm
baryons and in the light baryons [102]. Note that there are no 7/2 operators with nega-
tive parity that can be constructed purely with non-relativistic quarks, and the signal for
the extracted state is a result of the inclusion of the relativistic operators. We also able
to identify a few states which has strong overlap to hybrid operators. Though there are
more number of non-relativistic operators with hybrid nature in Table 3.6, we could not
clearly identify those. We expect a data with higher statistics will aid in extracting the
signals from those states. Further, one cannot argue an SU(6)⊗O(3) symmetry for the
higher lying states to these non-relativistic bands, because of same reasons as described
for triply charm baryons. Identification of the negative parity states with hybrid content
also suffers from this ambiguity.
Since both the doubly charm baryons have the same flavor structure, pqq, and hence
the same allowed flavor symmetry combinations, the interpolators used in their spec-
troscopy are similar. They differ only in their flavor content. Hence one expects a similar
pattern in their spectra also. To check such a similarity in the spectra of doubly charm
baryons, in Figure 4.10 and Figure 4.11, we show the extracted positive and negative
parity spectra, respectively, in terms of another type of energy splittings. For each spin,
left column is for splittings of Ξcc−D and the right column is for Ωcc−Ds splittings. By
subtracting out D and Ds mesons from Ξcc and Ωcc baryons we effectively leave only the
excitation of one charm quark in both cases. One would thus naively expect that both
spectra will be equivalent. This is almost true for the lowest state in each spin parity
channel, except for negative parity spin-7/2 state, as can be observed in Figure 4.10 and
Figure 4.11. That state is obtained from relativistic operators and it is expected that
such naive expectation may not hold there. This is also true for other excited states,
where the contribution from relativistic operators are much larger. The inset figures in
Figure 4.10 are for the positive parity ground states of spin-1/2 and spin-3/2 baryons.
Charm baryon spectrum 79
Ξcc
Ωcc
Ξcc
Ωcc
Ξcc
Ωcc
Ξcc
Ωcc
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
∆m
ass
(G
eV
)
1.7
1.72
1.74
1.76
1.78
1.8
1.781.791.81.811.821.831.841.851.86
1/2+
3/2+
5/2+
7/2+
(a)
(a)
(b)
(b)
Figure 4.10: Energy splittings of positive parity Ξcc and Ωcc baryons from D and Ds
mesons, respectively. For each spin, left column is for splittings of Ξcc(ccu) baryons fromthe ground state D(cu) meson and the right column is for splittings of Ωcc(ccs) baryonsfrom the ground state Ds(cs) meson. The inset figure are for the positive parity groundstates of spin-1/2 and spin-3/2 ground states. Results for D and Ds mesons are takenfrom Ref. [107].
Results for D and Ds mesons are taken from Ref. [107].
As mentioned in the introduction that the experimental discovery of doubly heavy
baryons is quite unsettled. It is thus important to compare the ground state spectra
of the two doubly heavy baryons obtained by different calculations. In Figure 4.12 we
show the ground state results of JP = 12
+and 3
2
+Ωcc baryons, obtained in this work,
along with other lattice as well as various model results. In Figure 4.13, we show the
ground state results of JP = 12
+and 3
2
+of Ξcc baryons, obtained in this work, along with
the only experimental (SELEX) result and other lattice as well as various model results.
The discretization scheme for charm quarks followed by the other lattice calculations
are as follows : Mathur, et. al [80]→NRQCD; Briceno, et. al [84] and PACS-CS [88]
→relativistic heavy quark formalism; ILGTI [127] → overlap fermions; and Bali, et.
al [86]→clover Wilson fermions. Our results are at pion mass 391 MeV, results for PACS-
CS [88] and Briceno et. al [84] are extrapolated to the physical pion mass, while ILGTI
[127] and Bali, et. al [86] results are at pion mass 340 MeV and 348 MeV respectively.
The lattice spacing in the temporal direction (at) for this work is 0.035 fm, the lattice
Charm baryon spectrum 80
Ξcc
Ωcc
Ξcc
Ωcc
Ξcc
Ωcc
Ξcc
Ωcc
1.95
2.10
2.25
2.40
2.55
2.70
2.85
3.00
3.15
∆m
ass
(G
eV
)
1/2-
3/2-
5/2-
7/2-
Figure 4.11: Energy splittings of negative parity Ξcc and Ωcc baryons from D and Ds
mesons, respectively. For each spin, left column is for splittings of Ξcc(ccu) baryons fromthe ground state D(cu) meson and the right column is for splittings of Ωcc(ccs) baryonsfrom the ground state Ds(cs) meson. Results for D and Ds mesons are taken fromRef. [107].
500
550
600
650
700
750
800
850
900
This
-work
PA
CS
-CS
-13
ILG
TI-
13
Bali-
12
Briceno-1
2
Math
ur-
02
RT
QM
-08
RQ
M-0
6
∆m
ass (
MeV
)
Ωccs (1/2+) − ηc
600
650
700
750
800
850
900
950
1000
This
-work
PA
CS
-CS
-13
ILG
TI-
13
Bali-
12
Briceno-1
2
Math
ur-
02
RT
QM
-08
RQ
M-0
6
∆m
ass (
MeV
)
Ωccs (3/2+) − ηc
Figure 4.12: Ground state masses of spin-1/2 and spin-3/2 doubly charm Ωcc baryonsas a splitting from ηc meson mass. Our results are shown by the red filled circle. Otherlattice as well as model results are also shown.
constants in other lattice calculations are as follows : PACS-CS at = 0.09 fm, Bali et.
al at = 0.08 fm and for ILGTI at = 0.0582 fm. Results from only Briceno et. al are
extrapolated to the continuum.
Charm baryon spectrum 81
400
500
600
700
800
900
Expt-
?
This
-work
PA
CS
-CS
-13
Bali-
12
Briceno-1
2
Math
ur-
02
RT
QM
-08
RQ
M-0
6
∆m
ass (
MeV
)
Ξccu (1/2+) − ηc
500
550
600
650
700
750
800
850
900
This
-work
PA
CS
-CS
-13
Bali-
12
Briceno-1
2
Math
ur-
02
RT
QM
-08
RQ
M-0
6
∆m
ass (
MeV
)
Ξccu (3/2+) − ηc
Figure 4.13: Ground state masses of spin-1/2 and spin-3/2 doubly-charmed Ξcc baryonsas a splitting from ηc meson mass. Our results are shown by the red filled circle. Our uquark mass is not physical and have a pion mass 391 MeV, while results for PACS-CS[88], and Briceno et. al [84] are extrapolated to the physical pion mass, while ILGTI[127] and Bali, et. al [86] results are at pion mass 340 MeV and 348 MeV respectively.
4.2.2 Investigating lattice artifacts
0.6
0.8
1.0
1.2
1.4
1.6
1.8
∆m
ass
(G
eV
)
0.7
0.72
0.74
0.76
0.78
0.80.75
0.8
0.85
0.9
cSW
=1.35 cSW
=1.35 cSW
=1.35 cSW
=1.35 cSW
=2.0cSW
=2.0cSW
=2.0cSW
=2.0
1/2+
3/2+
5/2+
7/2+
(a)
(a)
(b)
(b)
Figure 4.14: Comparison of a few low lying positive parity Ωcc baryon states from thetwo calculations; one using the tree level tadpole improved spatial clover co-efficient,cs = 1.35, and the other using ad-hoc boosted value, cs = 2.0, to study the systematicscoming from the discretization. The spectrum is in terms of the energy splittings frommηc in the respective calculations.
As in our previous study for triply charm baryons [112], to check the systematic
uncertainty due to spatial discretisation artifacts, we carried another calculation in which
the spatial clover co-efficient was boosted from its tree-level tadpole improved value,
Charm baryon spectrum 82
1.0
1.2
1.4
1.6
1.8
2.0
∆m
ass
(G
eV
)
cSW
=1.35 cSW
=1.35 cSW
=1.35 cSW
=1.35 cSW
=2.0cSW
=2.0cSW
=2.0cSW
=2.0
1/2-
3/2-
5/2-
7/2-
Figure 4.15: Comparison of a few low lying negative parity Ωcc baryon states from thetwo calculations; one using the tree level tadpole improved spatial clover co-efficient,cs = 1.35, and the other using ad-hoc boosted value, cs = 2.0, to study the systematicscoming from the discretization. The spectrum is in terms of the energy splittings frommηc in the respective calculations.
0.6
0.8
1.0
1.2
1.4
1.6
1.8
∆m
ass
(G
eV
)
0.6
0.65
0.7
0.75 0.7
0.75
0.8
0.85
cSW
=1.35 cSW
=1.35 cSW
=1.35 cSW
=1.35cSW
=2.0 cSW
=2.0 cSW
=2.0 cSW
=2.0
1/2+
3/2+
5/2+
7/2+
(a)
(a)
(b)
(b)
Figure 4.16: Comparison of a few low lying positive parity Ξcc baryon states from thetwo calculations; one using the tree level tadpole improved spatial clover co-efficient,cs = 1.35, and the other using ad-hoc boosted value, cs = 2.0, to study the systematicscoming from the discretization. The spectrum is in terms of the energy splittings frommηc in the respective calculations.
Charm baryon spectrum 83
1.0
1.2
1.4
1.6
1.8
2.0
∆m
ass
(G
eV
)
cSW
=1.35 cSW
=1.35 cSW
=1.35cSW
=2.0 cSW
=2.0 cSW
=2.0
1/2-
3/2-
5/2-
Figure 4.17: Comparison of a few low lying negative parity Ξcc baryon states from thetwo calculations; one using the tree level tadpole improved spatial clover co-efficient,cs = 1.35, and the other using ad-hoc boosted value, cs = 2.0, to study the systematicscoming from the discretization. The spectrum is in terms of the energy splittings frommηc in the respective calculations.
cs = 1.35 to cs = 2. At this boosted value the new extracted hyperfine splitting was
found to be physical [106]. The mass splittings were found to have increased by ∼45 MeV
between two calculations. For doubly charm baryon also we performed similar analysis
with the boosted spatial clover co-efficient. Figure 4.14, 4.15, 4.16 and 4.17 shows the
comparison of a few low lying doubly charm baryons using the two different charm quark
actions. We find a similar shift in the low lying doubly charm baryon energies, indicating
a similar scale of uncertainty in this calculation. While for the higher excited states this
shift seems to have eclipsed by the statistical uncertainties.
4.2.3 Energy splittings in doubly charm baryons
The energy splittings between various excitations in a spectra provide important informa-
tion about the nature of interactions needed to excite those states. The energy splittings
also provide inputs for building models which can describe these states successfully. The
most notable spin dependent baryon energy splitting that one always considers is the
hyperfine splittings between 32
+and 1
2
+states (for example, splitting between ∆ and
nucleon). For doubly charm baryons, we also compute this splitting and show in Figure
4.18 for Ξcc (left plot) and Ωcc (right plot) baryons. Our results are compared with other
Charm baryon spectrum 84
lattice results as well as with various model results. As one can see that our estimates
are in very good agreement with other lattice estimations.
0
50
100
150
200
Th
is-w
ork
PA
CS
-CS
-12
Ba
li-1
2
Bricen
o-1
2
ET
MC
-12
H.N
a-0
8
Ch
iu-0
5
Fly
nn
-03
Le
wis
-02
Ma
thu
r-0
2
QM
-08
RT
QM
-08
Bra
m-0
5
QC
DS
R-0
2
RQ
M-0
2
FH
T-0
2
HQ
ET
-94
∆m
ass (
Me
V)
Ξccu(3/2+) - Ξccu(1/2
+)
0
50
100
150
200
Th
is-w
ork
PA
CS
-CS
-13
ILG
TI-
13
Ba
li-1
2
Bricen
o-1
2
ET
MC
-12
H.N
a-0
8
Ch
iu-0
5
Fly
nn
-03
Le
wis
-02
Ma
thu
r-0
2
QM
-08
RT
QM
-08
QC
DS
R-0
2
RQ
M-0
2
FH
T-0
2
HQ
ET
-94
∆m
ass (
Me
V)
Ωccs(3/2+) - Ωccs(1/2
+)
Figure 4.18: Hyperfine mass splittings between spin-32
+and spin-1
2
+states of Ξcc (left
plot) and Ωcc (right plot) baryons are compared for various lattice and model results.Our results are shown by red filled circle. Pion mass in these calculations is 391 MeV,while results for PACS-CS [88], and Briceno et. al [84] are extrapolated to the physicalpion mass, while ILGTI [127] and Bali, et. al [86] results are at pion mass 340 MeV and348 MeV respectively.
In Figure 4.19, we compare these hyperfine splittings for Ξcc and Ωcc baryons as a
function of quark mass. In the x-axis of that figure, we use the square of the pseudoscalar
meson mass (mps), while y-axis shows hyperfine splittings at those pseudoscalar meson
masses. Along with Ξcc and Ωcc, we also show splittings between spin-32
and spin-12
states
of Ωccc baryon. The later splittings is not hyperfine in nature. For the Ωccc baryons, we
take the spin-orbit splittings between E3(32
+) and E0(1
2
+) states, which have same L and
S values corresponding to the 7th row in SF in Table 3.6.
In Figure 4.20, we show energy splittings of the ground states of each spin-parity
channel from the lowest state in that parity channel. For the positive parity, the lowest
state is JP = 12
+and for negative parity the lowest state is JP = 1
2
−. It is interesting
to note that both for Ξcc and Ωcc this splittings are almost same. This indicates that
the interquark interactions, which are responsible for these splittings, are similar in these
two different type of baryons. The negative parity 7/2 state is a result of the inclusion
of the relativistic operators and doesn’t appear in a purely non-relativistic spectrum
constructed using up to two derivatives. So the contribution to that state from a possible
three derivative operator is unclear and hence the disagreement may change after its
inclusion.
Motivated by the success of HQET inspired fit form in understanding the quark
Charm baryon spectrum 85
0
20
40
60
80
100
120
0.16 0.49 8.89
∆m
ass (
MeV
)
(ccu) (ccs) (ccc)mps2 (logscale)
3/2+ − 1/2
+
Figure 4.19: Mass splittings between spin-32
+and spin-1
2
+states are compared for
Ξcc(ccu), Ωcc(ccs) and Ωccc(ccc) baryons. For Ωccc, mass splitting is between E0(1/2+)and E3(3/2+) states, which is actually due to spin-orbit coupling, while for Ξcc and Ωcc,these are between respective ground states and are due to hyperfine splittings.
mass dependence of energy splittings in triple flavored baryons, we fit similar energy
splittings for doubly charm baryons. However, there is a difference in doubly charm
baryons. So as to show quark mass dependence one need to include triply-charmed
baryons also. In the study of triple flavored baryons, states at different quark masses
are obtained from same operators and so they are easy to compare. However, states at
doubly charmed-baryons and triply-charmed baryons are not obtained from same set of
operators except for positive parity spin 3/2 states. We thus consider energy splittings
only for this state. In specific, we calculate the following energy splittings : Ξ∗cc(ccu) −Du(cu) ,Ω∗cc(ccs)−Ds(cs) and Ω∗ccc(ccc)−ηc(cc), and Ξ∗cc(ccu)−D∗u(cu) ,Ω∗cc(ccs)−D∗s(cs)and Ω∗ccc(ccc)− J/ψ(cc) and plot in Figure 4.21. Absorbing the mass of the one remnant
valence charm quark in to the additive parameter,‘a’, we again fit these splittings with
a form a + b/mps, which was used for triply charm baryons, and obtain good fits. We
also like to point out that energy splittings at very light quark masses should not be
calculated using the above form and is valid only in the heavy quark domain.
It is interesting to note that one can extrapolate the fitted results to bottom mass to
obtain the energy splittings of Ω∗ccb(32
+)−Bc and Ω∗ccb(
32
+)−B∗c at the mass of Bc meson.
Using these splittings, we can empirically predict the mass splitting of B∗c −Bc by taking
the difference between Ω∗ccb(32
+)−Bc and Ω∗ccb(
32
+)−B∗c . Further, one can predict the mass
of Ω∗ccb by taking input for the mass of only discovered charm-bottom meson [16], assuming
it is as a pseudoscalar meson with mass 6277 MeV, using the splitting Ω∗ccb(32
+) − Bc.
Charm baryon spectrum 86
3/2+
5/2+
7/2+
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
∆m
ass
[G
eV
]
Ξcc
Ωcc
3/2-
5/2-
7/2-0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
∆m
ass
[G
eV
]
Ξcc
Ωcc
Figure 4.20: Energy splittings (in units of GeV) between the ground states of differentspins are shown for positive (top two) and negative parity (bottom two) doubly-charmedbaryons. Left two plots are for Ξcc and right two are for Ωcc baryons. For positive parity,the lowest state has JP = 1
2
+and for negative parity the lowest state has JP = 1
2
−.
Figure 4.21: Energy splittings between positive parity spin-3/2 baryons and pseudoscalarmeson as well as vector mesons are plotted against the square of the pseudoscalar masses.Left figure is the splittings of : Ξ∗cc(ccu)−Du(cu) ,Ω∗cc(ccs)−Ds(cs) and Ω∗ccc(ccc)−ηc(cc),and right figure includes following splittings : Ξ∗cc(ccu)−D∗u(cu) ,Ω∗cc(ccs)−D∗s(cs) andΩ∗ccc(ccc)− J/ψ(cc); they are plotted against the square of the pseudoscalar masses (i.e.,at Du , Ds) and ηc mass. We fit the quark mass dependence with a form a+ b/mps (rightfigure is also fitted with a constant term). The fitted results are shown by solid lines withshaded regions as one sigma error bars.
This way we obtain the mass splitting of B∗c − Bc as 80 ± 8 MeV (fitting with a form
a + b/mps) and 76 ± 7 MeV (fitting with a constant term). This result agrees very well
with the results obtained in potential models [162]. However, this result does not agree
with other lattice QCD prediction [74, 75, 163], which provided much lower values. It
will be interesting to study the lattice cut-off effect on the hyperfine splittings of these
hadrons with the possibility of future discovery of B∗c meson. By taking mBc = 6277
Charm baryon spectrum 87
MeV, our other prediction, the mass of Ω∗ccb(3/2+) is 8050 ± 10 MeV, which is consistent
with the results from various models [30, 34, 38, 50] and a recent lattice calculation [164].
4.3 Singly charm baryons
Singly charm baryons have received significant scientific attention, mainly due to the
recent experimental discoveries at various particle colliders like, SELEX, CDF, D0, Belle
etc. In contrast to the triply and doubly charm baryons, the production and identification
of singly charm baryons are relatively easier in hadron-hadron colliders as well as in
electron-positron colliders. Particularly, the new Beijing Spectrometer and Belle II has
great potential for accumulating large number of events and thus help us understand
more about single charm baryons. The PANDA experiment, a GSI future project, and
the LHCb are also expected to provide huge amount of information on charm baryon
spectrum.
Considering the experimental prospects, it is very timely to study the singly charm
baryons using a first principle non-perturbative calculation. That will serve various pur-
poses : understanding the structure of already discovered states, assigning quantum
numbers to ambiguous states and finally predicting states, which can be searched exper-
imentally. After triply and doubly charm baryons, we studied singly charm baryons with
the same detailing. However, for singly charm baryons the number of channels are many
more, namely, Λc(cud), Σc(cuu), Ξc(cus), and Ωc(css). In the presence of one or two light
quarks, the dynamics of these particles become much more complicated and hence they
provide a unique opportunity to study the interaction between one or more light quarks
in the presence of a heavy quark. These systems provide an excellent laboratory to study
the dynamics of the light quark in the heavy quark environment, such as those controlled
by the chiral symmetry. In a heavy quark effective theory, the two light quarks in the
singly charm baryon can often be considered as a diquark system, similar to the case
in doubly charm baryons. In the heavy quark limit, the spin-spin interactions between
the heavy quark and the diquark will vanish and hence the diquark spin will be a good
quantum number and will have implications in the singly charm spectrum as degeneracies
between the spin-spin coupled states.
Charm baryon spectrum 88
4.3.1 The spectrum
In this case, the number of possible interpolating operators are more than triply and dou-
bly charm baryons as was shown in Table 3.8. We considered a large set of interpolating
operators, as high as 180 (in Hg irrep for Ξc baryons), in construction of the correlation
functions. In the analysis, we considered the full set of relativistic and non-relativistic
operators for all the singly charm baryons, except for the Ξc, where we used only the
non-relativistic operators. We show our results for the spectra of singly charm baryons
in Figure 4.22, 4.23, 4.25 and 4.26, while the respective estimates are tabulated in the
Tables 4.5, 4.6, 4.7 and 4.8. As in the previous cases, we plot the spectra in terms of
the mass splittings instead of the absolute values. This is because the mass splittings
are expected to suffer less from the discretization errors, as they gets canceled between
the individual masses. In these plots, we show the mass splittings of these channels from
0.5 mηc , where the effects of charm quark discretization cancels out.
As in the previous cases, we observed that the pattern of the extracted low-lying
states are remarkably similar to the expectation from a model with broken SU(6)×O(3)
symmetry. As previously, the low lying bands that have relatively large overlap with
the non-relativistic operators are shown with magenta boxes. Similarly the states with
significant overlap with the operators proportional to the field strength tensor are painted
with thick borders, and are identified as strongly hybrid in nature.
12
+ 32
+ 52
+ 72
+ 12
− 32
− 52
− 72
−
1.072(10) 1.185(10) 1.827(21) 1.058(25) 1.443(12) 1.524(12) 1.522(22) 2.105(37)1.825(15) 1.825(20) 1.838(21) 1.822(16) 1.448(10) 1.533(10) 2.093(27)1.872(18) 1.825(40) 1.916(21) 1.612(13) 1.626(12) 2.083(38)1.883(25) 1.883(13) 1.950(29) 2.122(26) 2.092(19) 2.138(30)1.892(16) 1.904(20) 2.055(30) 2.165(31) 2.257(16)1.918(16) 1.943(20) 2.180(25) 2.261(28)2.022(19) 1.979(17) 2.193(28) 2.310(24)2.591(30) 2.027(23) 2.251(27)2.533(28) 2.033(18) 2.340(20)
2.511(35)
Table 4.5: Relativistic spectrum mΣc− 0.5 mηc in units of GeV.
Charm baryon spectrum 89
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
m -
0.5
mη
c
[G
eV
] Σ
c
Figure 4.22: Results on the spin identified spectra of Σc for both even and odd paritiesw.r.t. 0.5 mηc . The keys are same as in Figure 4.1.
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
m -
0.5
mη
c
[G
eV
]
Ωc
Figure 4.23: Results on the spin identified spectra of Ωc for both even and odd paritiesw.r.t. 0.5 mηc . The keys are same as in Figure 4.1.
Charm baryon spectrum 90
12
+ 32
+ 52
+ 72
+ 12
− 32
− 52
− 72
−
1.210( 6) 1.262(14) 1.927(21) 1.945(14) 1.515(34) 1.593(21) 1.674(19) 2.267(21)1.895(35) 1.849(43) 1.936(21) 2.184(18) 1.550(17) 1.619(19) 2.175(25)1.913(17) 1.910(30) 1.954(15) 1.744(12) 1.750(10) 2.210(32)1.964(18) 1.939(24) 2.112(24) 2.274(41) 2.157(33) 2.338(31)2.014(22) 2.045(28) 2.152(27) 2.280(20) 2.315(38) 2.381(28)2.046(25) 2.054(26) 2.305(17) 2.333(17) 2.417(31)2.059(74) 2.075(25) 2.360(19) 2.359(38)2.155(14) 2.079(27) 2.499(16) 2.421(20)2.258(90) 2.109(23) 2.426(40)
Table 4.6: Relativistic spectrum mΩc− 0.5 mηc in units of GeV.
4.3.2 Σc and Ωc baryon
The Σ++c and Ωc baryons have the same flavor structure : pqq. The allowed flavor
symmetries are similar to that of the doubly charm baryons, where one can have S flavor
structure and MS/MA flavor structure. Thus the spectrum of these two baryons should
resemble the spectrum obtained for the doubly charm baryons. This is quite evident from
the Figure 4.22 and Figure 4.23. In Figure 4.24, we show the hyperfine splittings of the
Ωc and Σc baryons in units of MeV. Alongside we plot the estimates from various lattice
calculations available in the literature and quark model calculations. We also show the
experimental value in the plot with a magenta filled polygon in each of the plots. We see a
reasonably good agreement between various lattice calculations, though our estimate for
the Σc hyperfine splitting lies above the experimentally observed value. One important
caveat in these calculations is that the light pion mass is heavier than its experimental
value and the disagreement between our estimate and the experimental value is mainly
due to this unphysical pion mass. The box size of the lattice used in these calculations is
L ∼ 1.9fm. The finite size effects in singly charm baryons can be expected to be larger
than the corresponding effects in triply and doubly charm baryons. Hence, a systematic
study of finite size effects is also necessary, which is beyond the scope of the current work.
4.3.3 Λc baryon
The Λc baryons are those with quark content udc and can have an A or MS/MA flavor
structure, which corresponds to Λc-singlet and Λc-octet respectively. The difference be-
tween the Λc-octet and the Σ+c is that the latter is a part of the isospin triplet, while the
Charm baryon spectrum 91
0
25
50
75
100
125
150
175
200
Expt.
This
-work
PA
CS
-CS
-13
ILG
TI-
13
Bali-
GT
-12
Bali-
HQ
ET
-12
Briceno-1
2
ET
MC
-12
Chiu
-05
Math
ur-
02
Lew
is-0
1
QM
-08
∆m
ass (
MeV
)Ωcss (3/2
+-1/2
+)
(a)
0
25
50
75
100
125
150
175
200
Expt.
This
-work
PA
CS
-CS
-13
Bali-
GT
-12
Bali-
HQ
ET
-12
Briceno-1
2
ET
MC
-12
Chiu
-05
Math
ur-
02
Lew
is-0
1
QM
-08
∆m
ass (
MeV
)
Σcuu (3/2+-1/2
+)
(b)
Figure 4.24: Hyperfine splittings in Ωc and Σc baryons(in MeV). The estimates from thiscalculation has been quoted in red filled circles, while estimates from other lattice calcula-tions in black squares, and a quark model calculation in blue square. The experimentallyobserved value is quoted as magenta filled polygon. Pion mass in these calculations is391 MeV, while results for PACS-CS [88], and Briceno et. al [84] are extrapolated to thephysical pion mass, while ILGTI [127] and Bali, et. al [86] results are at pion mass 340MeV and 348 MeV respectively.
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-
7/2-0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
m -
0.5
mη
c
[G
eV
]
Λc
Figure 4.25: Results on the spin identified spectra of Λc for both even and odd paritiesw.r.t. 0.5 mηc . The keys are same as in Figure 4.1.
Charm baryon spectrum 92
12
+ 32
+ 52
+ 72
+ 12
− 32
− 52
− 72
−
0.969(11) 1.732(19) 1.685(31) 1.995(25) 1.353(14) 1.410(20) 1.706(22) 2.156(34)1.761(19) 1.811(70) 1.934(40) 1.517(22) 1.541(31) 1.950(39) 2.292(42)1.825(28) 1.923(16) 2.017(29) 1.518(12) 1.698(18) 2.094(40)1.837(33) 1.927(25) 2.061(39) 2.084(66) 2.073(59) 2.397(29)1.926(17) 2.047(22) 2.148(21) 2.220(21) 2.224(19)1.981(23) 2.058(25) 2.239(26) 2.244(18)2.067(27) 2.128(24) 2.287(41) 2.258(45)2.272(55) 2.371(26) 2.298(19) 2.265(41)2.295(51) 2.568(32) 2.320(23) 2.354(43)2.403(41) 2.480(28) 2.419(18)2.489(30)2.669(23)
Table 4.7: Relativistic spectrum mΛc− 0.5 mηc in units of GeV.
former forms an isospin singlet. The details of the flavor structure is shown in Table 3.2.
Since there is no totally antisymmetric spin combination that can be constructed from
a purely non-relativistic formalism, one requires angular momentum structure through
non-local behavior to construct a flavor-singlet Λc interpolating operator. Figure 4.25
shows the spectrum of Λc, where one can observe only one state in the low lying band.
This state has significant overlap with the flavors MS/MA with non-relativistic spin 1/2.
As observed for the other channels, the numbers and the pattern of the first negative
parity band and the first excited positive parity band agrees with the non-relativistic
predictions. It is to be noted that the energy ordering of the low lying two positive parity
states are in accordance with the experimental observation.
4.3.4 Ξc baryon
The Ξc baryons, with quark content usc, can have S, A or MS/MA flavor structure,
which correspond to Ξc-decuplet, Ξc-singlet and Ξc-octet respectively. Within Ξc-octet
flavor structure, one has two possibilities. If we assume an us-spin symmetry, similar to
the isospin that describes the symmetry between u and d quarks, there are two possible
MS/MA combinations : the Ξc-octet and the Ξ′c-octet. The latter is a part of the us-spin
triplet, of which Ωc and Σ++c are included, while the former is a us-spin singlet. As in
the case of Λc, due to the absence of a totally antisymmetric spin combination that can
be constructed from a purely non-relativistic formalism, one requires angular momentum
Charm baryon spectrum 93
1/2+
3/2+
5/2+
7/2+
1/2-
3/2-
5/2-1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
m -
0.5
mη
c
[G
eV
] Ξ
c
Figure 4.26: Results on the spin identified spectra of Ξc for both even and odd paritiesw.r.t. 0.5 mηc . The keys are same as in Figure 4.1.
0
25
50
75
100
125
150
175
200
Expt.
Th
is-w
ork
PA
CS
-CS
-13
Bali-
GT
-12
Bali-
HQ
ET
-12
Briceno
-12
ET
MC
-12
Math
ur-
02
Lew
is-0
1
QM
-08
∆m
ass (
MeV
)
Ξcsu (3/2+-1/2
+)
Figure 4.27: Hyperfine splittings in Ξc baryon(in MeV). The estimates from this calcula-tion has been quoted in red filled circles, while estimates from other lattice calculationsin black squares, and a quark model calculation in blue square. The experimentallyobserved value is quoted as magenta filled polygon.
structure through non-local behavior to construct a flavor-singlet Ξc interpolating oper-
ator. Hence the low lying spectra for Ξc baryon will have two spin 1/2 states from the
Charm baryon spectrum 94
12
+ 32
+ 52
+ 72
+ 12
− 32
− 52
− 72
−
1.033(12) 1.221(10) 1.856(16) 1.985(14) 1.467(15) 1.458(26) 1.640(11)1.148(10) 1.805(28) 1.930(19) 2.019(22) 1.504(26) 1.572(43) 1.758(13)1.738(55) 1.845(21) 1.935(17) 2.180(22) 1.528(33) 1.608(38)1.762(14) 1.870(36) 2.019(19) 1.563(13) 1.619(22)1.923(18) 1.917(30) 2.030(18) 1.585(13) 1.622(38)1.931(19) 1.953(18) 2.089(21) 1.692(22) 1.679(40)1.957(18) 2.037(17) 2.147(24)1.982(19) 2.048(15) 2.164(21)2.043(21) 2.059(40) 2.198(16)2.074(24) 2.037(32) 2.202(15)2.078(25) 2.079(19) 2.670(90)2.083(17) 2.116(23)2.126(19) 2.125(20)2.138(21) 2.128(23)2.280(42) 2.146(24)2.437(56) 2.162(54)2.517(35) 2.721(51)2.551(34) 2.771(55)2.640(37)2.738(42)
Table 4.8: Relativistic spectrum mΛc− 0.5 mηc in units of GeV.
Ξc-octet and Ξ′c-octet operators and a spin 3/2 state with the Ξc-decuplet flavor struc-
ture. This can clearly be observed in Figure 4.26. We have calculated the correlation
matrix for all of them. However, being too many in numbers, it is quite difficult to put
together all these operators in our framework of variational fitting code. Hence, in this
work, we use a smaller set comprising only the non-relativistic operators for fitting, and
the Ξc spectrum shown in Figure 4.26 is non-relativistic. In Figure 4.27, we show the
hyperfine splitting between the 3/2+ and 1/2+ ground states of Ξc baryon. Alongside
we show the estimates from other lattice calculations and quark model estimate. The
experimental value is shown as magenta filled polygon.
4.3.5 Energy splittings : A preliminary study
One important caveat in these calculations of the excited state charm baryon spectroscopy
is that the pion mass used, mπ = 391 MeV. This unphysical heavy pion mass can give
serious implications in the spectrum, where one expects the light quark chiral dynamics
Charm baryon spectrum 95
Λc
Σc
Ξc
Ωc
Λc
Σc
Ξc
Ωc
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
m -
m1
- [G
eV
]
1/2+
1/2-
⊗⊕
⊕⊗
⊗⊕
⊕⊗ ⊕⊗
⊕ ?
⊕ ?
⊕ ?⊗
⊗
⊗
Figure 4.28: Spectra of spin 1/2 singly charm baryons shown as their splitting from thelight vector mesons, such that the effective valence content in the splitting is a charmquark. Explicitly we consider the following splittings : Λc(cud)−ρ(ud), Σc(cuu)−ρ(uu),Ξc(cus)−K∗(us) and Ωc(css)− φ(ss).
to play a significant role. So as to make a comparative study of light quark effects in the
singly charm baryons, we consider the splitting of the singly charm baryons with respect
to the corresponding light vector mesons, such that there is only one remnant valence
charm quark in the splitting. In other words, we consider the splittings, Λc(cud)−ρ(ud),
Σc(cuu)− ρ(uu), Ξc(cus)−K∗(us) and Ωc(css)− φ(ss). This is a good choice, since to
the leading order all these splittings should have the same discretization errors and the
same effect of the light quark dynamics, making a comparative study between the singly
charm baryon spectra possible. In Figure 4.28, 4.29, 4.30 and 4.31, we plot the spectra of
the singly charm baryons, as the splittings mentioned above, corresponding to the spin
1/2, 3/2, 5/2 and 7/2 respectively. We also plot the experimental values (violet circled
star) of the low lying singly charm states in these plots. Question marks are given beside
those states that for which the quantum numbers are not certain (even based on quark
model assumptions). It is very clear that the pattern of the low lying states, particularly
for 1/2+ and 3/2+ channels, is almost the same between the experimental estimates and
the results from our calculations. The little disagreement in the low lying spectra with
the experimental numbers could be because of the various systematics that we will be
Charm baryon spectrum 96
Λc
Σc
Ξc
Ωc
Λc
Σc
Ξc
Ωc
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6m
- m
1- [G
eV
] 3/2
+
3/2-
⊕ ?⊕ ?
⊕
⊗
⊕
⊗
⊗ ⊗ ⊗⊕
Figure 4.29: Spectra of spin 3/2 singly charm baryons shown in terms of the splittings :Λc(cud)− ρ(ud), Σc(cuu)− ρ(uu), Ξc(cus)−K∗(us) and Ωc(css)− φ(ss).
discussing in Section 4.4. This observation calls for a detailed study of the singly charm
baryon spectra, with improved control over the systematics, particularly with lighter pion
mass as well as with multi-hadron operators.
4.4 Caveats in excited state spectroscopy
Lattice computations should be accompanied with a study of the systematics associated
with the formulation of the theory on lattice, so as to validate and compare the results
obtained from these calculations with the actual values. The discretization of the space-
time introduces a UV cut off proportional to the inverse of the lattice spacing (1/a). The
systematics from this UV cut off is in general quantified in terms of the dimensionless
quantity, ‘am’, m being the quark mass. One in general considers complicated lattice
formalisms, that has the discretization errors up to different orders of ‘ma’ removed. In
this study, in order to have good control over the discretization errors, we use the tree
level tadpole improved (mean field improvement [111]) clover fermion action, which has
no O(mat) and O(mas) errors. We have not considered any higher order terms as the
temporal lattice spacing is quite small. However, for the charm quarks the discretization
Charm baryon spectrum 97
Λc
Σc
Ξc
Ωc
Λc
Σc
Ξc
Ωc
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
m -
m1
- [G
eV
]
⊗⊕
5/2+
5/2-
Figure 4.30: Spectra of spin 5/2 singly charm baryons shown in terms of the splittings :Λc(cud)− ρ(ud), Σc(cuu)− ρ(uu), Ξc(cus)−K∗(us) and Ωc(css)− φ(ss).
Λc
Σc
Ξc
Ωc
Λc
Σc
Ξc
Ωc
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
m -
m1
- [G
eV
]
7/2+
7/2-
Figure 4.31: Spectra of spin 7/2 singly charm baryons shown in terms of the splittings :Λc(cud)− ρ(ud), Σc(cuu)− ρ(uu), Ξc(cus)−K∗(us) and Ωc(css)− φ(ss).
Charm baryon spectrum 98
errors is significantly larger than the light and strange quarks on a given lattice. The
study of heavy hadrons using lattice QCD has this inherent problem, since at these
masses, with currently available lattices, the condition mat << 1 and mas << 1, in
general is not satisfied leading to larger systematic errors. In such a case, one require
to perform a continuum extrapolation of the estimates from calculations using multiple
lattices with very fine lattice spacings : this is beyond the scope in these excited state
spectroscopy calculations.
Another systematics that enter the lattice calculations is due to the finite box size,
which is limited by the available computational resources. To ensure negligible finite size
effects in these systems, in general one makes sure that the lattice size, L, is at least 4
times the Compton wavelength of the lightest state in that lattice. In our calculations,
mπL = 3.77 and hence we expect the finite size effects to be under control. An infinite
volume extrapolation is necessary to have the estimates free of finite size effects : This is
also beyond the scope of the previous work, as excited baryon spectroscopy calculations
involving larger lattices calls for very large computational requirements, which is currently
beyond the limit of the available resources.
A third systematics in lattice QCD calculations is related to the light quark masses
that are used in the calculations. The computational cost for calculations involving
fermions increases as the quark mass decreases and hence the studies involving lighter
quark masses are highly CPU intensive. The light quark mass used in these calculations
corresponds to a pion mass of ∼ 391 MeV, which is greater than the physical pion mass.
Hence, along with the continuum and the infinite volume extrapolations, one need to
perform a chiral extrapolation, which is not possible with just one light quark mass
available in these calculations. With all these caveats related to the lack of study of
systematics in mind, confronting this calculation with another non-perturbative study
with better control over the systematics, will be a good check to validate the smallness
of the systematic errors in our results from calculations in the previous chapters.
Finally we have not used the multi-hadron operators in our basis of operators. These
operators couples efficiently onto multi-hadron scattering states having the same quan-
tum number as the resonance states. As an example, while studying ρ resonance, one
needs to include ππ two-particle operators along with regular 1−− operator set. To match
the appropriate quantum number one use the technique of moving frames where the total
momentum of the system is nonzero. This has been demonstrated in Ref. [166], where ρ
resonance has been studied in great details. This is also true for ∆, S11(1535) resonance,
where one needs to use Nπ operators to study these resonances. Other higher particle
Charm baryon spectrum 99
decay channel operators also need to be included for comprehensive study of resonances.
These additional operators will increase the number of levels that can be extracted and
hence the resonance parameters can be evaluated from the extensive mapping of the en-
ergy dependence of the scattering amplitudes. In this calculation, one should also use
these multi-hadron operators in the basis of operators. Inclusion of those operators, par-
ticularly those involving light quarks, may affect some of the above conclusions. However,
due to the presence heavier quark(s), influence of these multi-hadron operators may be
lesser for charm baryons, particularly for triply and doubly charm baryons, than their
influence in the light baryons.
4.5 Summary, conclusions and future prospects
In this work, we present results from the first non-perturbative calculation on the excited
state spectroscopy of the charm baryons, with three, two and one valence charm quarks
and with spin up to 7/2. Employing a large set of operators classified according to the
irreps of the lattice symmetries, constructed out of derivative based operator formalism
using up to two derivatives, we extracted approximately 20 states for each of the charm
baryons, along with a reliable identification of the spin-parity quantum numbers for each
of these states. Beside identifying the spin of a state we are also able to decode the
structure of operators leading to that state : whether constructed by relativistic, non-
relativistic, hybrids, non-hybrid types or a mixture of them all. We discussed the results
in three sections, dedicated for triply, doubly and singly charm baryons respectively.
The spectra for the triply, doubly and singly charm baryons are shown in Figure 4.1,
4.9, 4.8, 4.22, 4.23, 4.25 and 4.26. Similar to light and strange baryon spectra [102], we
also find the number of extracted states of each spin in the three lowest-energy bands
and the number of quantum numbers expected based on weakly broken SU(6) × O(3)
symmetry agree perfectly, i.e., all the charm baryon spectra remarkably resemble the
expectations of quantum numbers from non-relativistic quark model [28, 29, 165]. It is
to be noted that in these calculations, we use the full set of operators (non-relativistic
and relativistic) and still the spectra behaves to be non-relativistic.
Discretization errors are of major concern for heavy hadron spectroscopy. Since a
calculation involving multiple lattice spacing is beyond the scope of this work, we follow
a similar analysis procedure, as was performed in Ref. [106]. To assess if radiative
corrections to the co-efficient of the improvement term in the charm quark action could
lead to significant changes in physical predictions, we perform a second calculation using
Charm baryon spectrum 100
a boosted spatial clover co-efficient, cs = 2, from its tree level tadpole improved value,
cs = 1.35. A very similar shift of ∼ 45 MeV, to that found in charmonia, was observed
for the lowest few states of triply and doubly charm baryons, indicating a similar scale
of uncertainty in this calculation of charm baryons also. The higher lying charm baryon
states do not show any statistically significant shifts between the estimates from the two
calculations.
Various energy splittings, including splittings due to hyperfine and spin-orbit coupling,
are also evaluated for all these baryons. In the triply charm baryons we performed a
quantitative study of the spin-orbit coupling to understand how good is the heavy quark
approximation for the charm quarks, by comparing them with the respective splitting for
triply light, strange and bottom baryons. We find degeneracy between these spin-orbit
split states is more or less satisfied both for bottom and charm quarks. However, data
with higher statistics is necessary to identify the breaking of this degeneracy precisely
at charm quark. The energy splitting of the triply charm baryon spectrum, from the
isoscalar vector meson (irrep T−−1 ) ground state is also evaluated. These splittings are
compared with similar ones obtained at other quark masses. For the splitting, which
mimics the binding energy of these states, significant quark mass dependence is observed
for ground as well as for first few excited states, except for the JP = 3/2+ and JP = 1/2−
ground states. These splittings can be modeled with a form a + b/mps to show their
expected quark mass dependence, which assumes they will tend to a constant in the
heavy quark limit. It is interesting to note this form gives a good fit with data at
bottom, charm as well as strange quark masses. For some of them, we observe that the
extrapolated fit lines pass through the light quark data points even through they are not
included in the fit.
For doubly and singly charm baryons, we calculated hyperfine mass splittings between
spin-32
+and spin-1
2
+states and compared those with various other lattice as well as
model results. We also evaluated mass splittings between ground states of spin-32
and
spin-12
states at both parities. To study the quark mass dependence of these splittings
we compared results for Ξcc(ccu), Ωcc(ccs) and Ωccc(ccc) baryons for which there is a
common ‘cc’ diquark and a varying quark from ‘u’ to ‘c’. Encouraged by a successful
fitting of the mass splittings in triple flavored baryons, we studied similar mass splittings.
Here also we find that a heavy quark motivated form a+ b/mps can fit quite successfully
energy splittings like : Ξ∗cc(ccu) −Du(cu) ,Ω∗cc(ccs) −Ds(cs) and Ω∗ccc(ccc) − ηc(cc), and
Ξ∗cc(ccu)−D∗u(cu) ,Ω∗cc(ccs)−D∗s(cs) and Ω∗ccc(ccc)− J/ψ(cc). From the fitted results we
are able to predict B∗c −Bc = 80± 8 MeV and Ω∗ccb(3/2+) = 8050 ± 10 MeV. The study
Charm baryon spectrum 101
of energy splittings of singly charm baryons is ongoing and we expect similar interesting
results from that sector also. The method used here to study charm baryons can also
be employed for similar study of bottom baryons. Considering experimental importance
and prospects we will carry out such calculation in future.
There are various systematics that are unattended in this calculation, which include
various extrapolations like chiral extrapolation, continuum limit and infinite volume ex-
trapolation. Furthermore, it is also to be noted that we have not used any multi-hadron
operators, which are expected to give significant implications in the extracted spectra.
Hence, one also need to perform calculations considering multi-hadron operators. How-
ever, this calculation is the first of its kind in the excited charm baryon spectra calculation
and hence this study serves as a foundation for many follow up studies, which may help
us address many of the challenges in the heavy baryon spectroscopy.
Chapter 5
Spectroscopy using chiral fermions
In this chapter, we discuss another independent non-perturbative calculation to deter-
mine the low lying charm baryon spectrum with improved control over the systematics.
This alternative calculation was made with a hope that confronting our ground state
studies with the excited state calculations will give us an idea of the systematic errors
appearing in our excited state spectrum calculations. We give a brief introduction of this
calculation in Section 5.1 and Section 5.2. As mentioned earlier, discretization errors are
of major concern in charm baryon spectroscopy. In Section 5.3, we detail our attempts
to quantify the discretization errors that enters in these calculations, by studying the
dispersion relation for the pseudoscalar charmonia. In Section 5.4, we discuss our results
on charmonia, charm-strange mesons and charm baryons. There we make comparison
of estimates from this studies with the experimental estimates, wherever available. We
also make comparison of estimates from this calculation and our studies of excited state
spectroscopy, wherever available. Finally, in Section 5.5, we summarize the findings in
these calculations.
5.1 Mixed action formalism
In order to study the systematics due to the discretization errors in our previous study, we
thus performed another calculation with a chiral action with the overlap fermions [123,
124], which is automatically O(a) improved. However, using overlap action for the dy-
namical quarks is still prohibitively costly, except with fixed topology [167]. A substan-
tially less expensive approach is to adopt a mixed action formulation with chiral fermions
as valence quarks in a background of existing highly improved gauge field configurations.
102
Spectroscopy using chiral fermions 103
Mixed action approaches have been studied by many groups such as DWF valence on
staggered fermion sea [63, 115, 116, 117], overlap valence on DWF sea [118], overlap
valence on clover sea [119], and overlap valence on twisted fermion sea [120]. The Indian
Lattice Gauge Theory Initiative (ILGTI) has adopted this mixed action approach and
this part of my thesis is related to my work in that project. This calculation uses a large
set of dynamical configurations generated during the past few years by MILC [122] with
the one-loop, tadpole improved Symanzik gauge action and highly improved staggered
quark (HISQ) fermion action [121]. Two degenerate light quarks, a strange and a charm
sea quark field were included in the generation of these dynamical configurations. With
very small magnitude of taste violations in comparison with the unimproved staggered
quarks, HISQ provides an efficient lattice fermion formulation that demands very less
computational requirements in contrast with the overlap action in simulation.
The valence quarks in this calculation are realized using the overlap formalism [123,
124], which has many desirable features. The overlap formalism has exact chiral symme-
try [124] on the lattice and is automatically O(a) improved. By adopting such a mixed
action approach, one can get advantage of the chiral symmetry and low quark mass limit
of overlap fermions, and the advantage of having a large set of configurations with small
discretization errors as well as small taste breaking effects. One also gets the advantage
of simulating both light, strange as well as heavy fermions on the same lattice formalism
with chiral fermions having no O(a) errors. The overlap action also has some desirable
features computationally, such as the adaptation of multi mass algorithms [125]. How-
ever, we will have to encounter the usual systematics related to mixed action and partial
quenching [169, 170, 171]. With the above formulation we have calculated the ground
state spectra of various charm hadrons and have made a comparative study between these
two projects along with other results from literature. Agreement between these results
from different approaches gives confidence in our previous estimates. The results from
these studies were reported in the Ref. [127, 126]
5.2 Numerical details
In this calculation, we used two sets of dynamical 2+1+1 flavors HISQ lattice ensembles,
generated by the MILC [122] : 323 × 96 lattices with lattice spacing a = 0.0877(10) fm
and 483 × 144 lattices with lattice spacing a = 0.0582(5) fm. The gauge action used in
these calculations is a one-loop, tadpole improved Symanzik gauge action that included
1× 1 and 1× 2 planar Wilson loops and a 1× 1× 1 ‘parallelogram’. The co-efficients of
Spectroscopy using chiral fermions 104
these terms are calculated perturbatively and are tadpole improved. As mentioned above,
the sea quark fields were realized using the HISQ action, which ensured the quark action
is order a2 improved. The use of HISQ action, instead the naıve staggered quark ac-
tion, ensured very small taste-symmetry violations. The lattice spacings of these lattices
mentioned above were determined by equating the Ω baryon mass measured on these en-
sembles to its physical value. Our estimates for the lattice spacings, as mentioned above,
are consistent with the estimates as measured by MILC using the r1 parameter [122],
which are a = 0.0888(8) fm and a = 0.0582(4) fm respectively. The strange and charm
masses are set at their physical values, while ml/ms = 1/5 for both lattices. The details
of these configurations are summarized in Ref. [122]. The results reported here were ob-
tained from 110 configurations on the coarser lattice, and 65 configurations on the finer
lattice.
For valence quarks we used overlap action [124]. The overlap Dirac operator has the
following definition for massive fermions :
Dov = (ρ+m
2) + (ρ− m
2)γ5sgn(γ5Dw); sgn(γ5Dw) =
γ5Dw√γ5Dwγ5Dw
, (5.1)
where sgn denotes the matrix sign function, Dw is the Wilson Dirac fermion operator
and ρ = 1/2κ− 4 here is a negative mass parameter. Luscher showed that on the lattice
one can define the chiral transformation as [147],
δψ = iαγ5(1− a
2Dov)ψ & δψ = iαψ(1− a
2Dov)γ5, (5.2)
which reduces to the proper continuum form in the continuum limit. For any a 6= 0,
the overlap action remains invariant under this lattice chiral transformation. While
the Wilson fermion action is ultra-local having only up to second derivative terms in
it, in overlap fermions the condition of ultra-locality is sacrificed to preserve the exact
chiral symmetry on the lattice in accordance with the Nielson-Ninomiya theorem. The
advantage of the overlap fermion over Wilson fermion is the presence of the exact chiral
symmetry. But since the operator contains a matrix sign function, its numerical studies
require more computational resources than the Wilson fermion formulation.
Hence, for the numerical implementation of massive overlap fermions the methods
used by the χQCD collaboration [168, 172] were followed. The gauge configurations were
first fixed to Coulomb gauge and then HYP smeared [173]. Low mode deflation technique
was employed in the inversion of the Hermitian Wilson Dirac operator (γ5Dw). In these
Spectroscopy using chiral fermions 105
calculations, we projected out 350 and 160 low lying Wilson eigenmodes for the coarser
and the finer lattices, respectively, using Arnoldi algorithm. The matrix sign function
of the Hermitian Wilson Dirac operator in the overlap Dirac operator is calculated with
22nd degree Zolotarev rational polynomial approximation [174, 175]. We used periodic
boundary condition in the spatial and anti-periodic in the temporal directions. Using
both point and wall sources and sinks, we calculated various hadron two point correlation
functions. The adaptation of the multi mass algorithms on overlap action helped us
to calculate the quark propagators over a wide range of quark masses with 10 − 12%
overhead. Our extracted pseudoscalar meson masses are within the range 400 − 5130
MeV and 230 − 4000 MeV for the coarser and finer lattices respectively. The valence
strange quark mass was tuned by setting the ss pseudoscalar mass to 685 MeV [176]. At
heavy quark masses, one need to be careful about the discretization errors. The charm
mass is tuned by setting the spin-averaged 1S state mass, (mηc +3mJ/ψ)/4, to its physical
value, where we take into account the kinetic mass, as defined below, in the definition
of mass. As in the previous sections, in the following sections also, we will discuss our
estimates for mesons and baryons mostly in terms of energy splittings as those have less
systematics compared to extracted energies.
5.3 Discretization errors
Since ma is not very small, we need to be careful about discretization errors. Overlap
action does not have O(ma) errors. In order to estimate the size of discretization errors
coming from higher orders of ma, we look at the energy-momentum dispersion relation of
the 1S charmonia. A naive construction of finite-momentum meson correlation functions
by projecting the point-point meson correlation function to a suitable momentum is
in general noisy, and require large statistics. Hence in these calculations of dispersion
relations, we use a momentum induced wall source to construct the quark propagators,
which are later used to project the meson correlators to suitable momentum. Below we
briefly discuss this method, which is found to improve the SNR ratio significantly in the
finite momentum correlators with much less statistics [127].
Spectroscopy using chiral fermions 106
5.3.1 Finite momentum meson correlators
The mode expansion of a quark propagator from a point source is given by
S(x′µ;xµ) =1
L3 Nt
∑kµ
eikµ(xµ−x′µ)S(kµ). (5.3)
A quark propagator constructed out of a momentum (k1,m = 2πmL
) induced wall source is
defined as :
∑x′1,x
′2,x′3
eik1,mx′1S(x′µ;xµ) =∑
x′1,x′2,x′3
eik1,mx′1
L3 Nt
∑kµ
eikµ(xµ−x′µ)S(kµ),
=eik1,mx1
Nt
∑k4
eik4(x4−x′4)S(k1,m, 0, 0, k4),
= S(x1, x4;x′4; k1,m). (5.4)
With m = 0, we get back the wall source,
∑x′1,x
′2,x′3
S(x′µ;xµ) =1
Nt
∑k4
eik4(x4−x′4)S(0, 0, 0, k4) = S(x4;x′4). (5.5)
Construction of finite momentum meson correlation function follows by introducing
another phase factor in the final momentum projection, so as to compensate the phase
factor that was introduced in the source. Local meson interpolating operators possess
the form ψΓψ(x ), where Γ carries the quantum numbers. With the use of momentum
induced wall sources, the finite momentum correlation functions for such operators can
be written as
∑x,y,z
e−ik1,nx1〈ψΓψ(x ) ¯ψmΓ′ψ0(x ′)〉
=∑x,y,t
TrΓS(x1, x4;x′4; k1,m)Γγ5S†(x4;x′4)γ5 e−ik1,nx1
= Tr Γ
Nt
∑k4
eik4(x4−x′4)S(k1,m, 0, 0, k4)
×Γγ5
Nt
∑k′4
e−ik′4(x4−x′4)S(0, 0, 0, k′4)γ5
∑x1
ei(k1,m−k1,n)x1 , (5.6)
Spectroscopy using chiral fermions 107
and ∑x,y,z
e−ik1,nx1〈ψΓψ(x ) ¯ψmΓ′ψ0(x ′)〉 =
M(k1,m) if m = n
0 if m 6= n(5.7)
Here ψm(x′) is the momentum induced quark source field with a phase factor, eik1,mx′1 ,
and M(k1,m) is the finite momentum meson propagator with a momentum, k1,m.
1.35
1.36
1.37
1.38
1.39
1.4
1.41
1.42
1.43
1.44
1.45
0 0.5 1 1.5 2
E(p)
p × 2π/L
point
MI wall
Figure 5.1: Energy-momentum dispersion relation for the ηc on the coarser lattices witha = 0.09 fm. Also shown are the continuum relativistic dispersion relation (solid line),and the lattice dispersion relation (dashed line) for the standard scalar action.
Employing this method and simple point source, we calculated pseudoscalar meson
mass at various external momenta p2 = n2 (2π/L)2, with n ≤ 2. While the momen-
tum induced wall requires a separate inversion for each momenta, that is more than
compensated by the improvement in the signal, as shown in Figure 5.1. In this fig-
ure, we show both point source and momentum induced wall source results for the
coarser lattice. Also shown are the continuum dispersion relation, E2 = m2 + p2,
and the lattice dispersion relation for the standard scalar action with O(m2a2) error,
sinh2(E(p)a/2) = (sinh2(ma/2))2 + sin2(pa/2). Note that the wall source results are
from 8 configurations, but the statistical error is already comparable to that from 100
configurations with the point source. While substantial deviation from the continuum
dispersion relation is seen, the lattice scalar dispersion relation seems to explain the data
quite well.
For a more quantitative analysis, we follow the standard practice of introducing an
effective “speed-of-light” c through E2(p) = m2 + p2c2. The value of c obtained using
Spectroscopy using chiral fermions 108
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5
c2
p × 2π/L
pointMI wall
Figure 5.2: c2 = (E2(p) − E2(p = 0))/p2 calculated for ηc at various values of p, onlattices with a = 0.09 fm. Solid horizontal lines represent the estimate ma/ sinh(ma),suggested by the lattice scalar action.
this relation and E(p) at various p are shown in Figure 5.2. As the figure shows, we get
c2 ∼ 0.75 on our coarser lattices. Using quenched overlap fermions, on an even coarser
lattice, Ref. [177] found c2 ∼ 1 for the charm. However, on our dynamical lattices, we
obtained c2 values similar to that obtained in the literature with clover action, indicating
a similar size of cutoff in the overlap action as that in the clover. The c2 values are
well-approximated by the estimate ma/ sinh(ma), suggested by the lattice scalar action.
With such a large deviation of c from unity, the discretization errors entering due to
uncertainty in the tuning of charm mass could be very large. In a second attempt, we
used an expansion of the energy-momentum dispersion relation in powers of pa, assuming
p << m0 & 1/a,
E(p)2 = M21 +
M1
M2
p2 +O(p4) = M21 + p2c2. (5.8)
Here M1 is the called pole mass, E(0) : which is the same as rest mass in our previous
definition, and M2 is called the kinetic mass (M1/c2). The difference between M1 and
M2 is a measure of O(ma) cutoff effects. As highlighted in Ref. [178], in the so-called
Fermilab interpretation, since M2 controls the non-trivial physics of a heavy hadron
system, in using a relativistic action for heavy quarks, one should use M2 to measure the
masses. In Figure 5.3, we show E(p)2 for various momenta for the pseudoscalar meson on
our finer lattices. The green line is for the continuum dispersion relation, E2 = m2 + p2,
while the blue line is the fitted dispersion relation with c = 0.96(2). For coarser lattices
Spectroscopy using chiral fermions 109
0.66
0.67
0.68
0.69
0.7
0.71
0.72
0.73
0.74
0.75
0.76
0 0.02 0.04 0.06 0.08 0.1
E(p)2
p2
c = 0.964
c = 1.0
Figure 5.3: Energy-momentum dispersion relation for the pseudoscalar meson at charmmass on the finer lattices. Blue line is with c = 0.96(2) obtained by fitting our data whilegreen line is with c = 1.
we obtain c = 0.92(3). With a better estimate for c, kinetic mass proved to be a good
candidate to be used for tuning the charm quark mass, as mentioned in the previous
section.
5.4 Results
In this section, we discuss our results from these calculations. We studied pseudoscalar
meson masses in a broad range of energies spanning between 400− 5130 MeV and 230−4000 MeV for the coarser and finer lattices respectively. Figure 5.4 shows two plots of
mπ versus mq for the range of quark masses we performed the computations for coarser
and finer lattices respectively.
We start our discussion with various energy splittings in mesons, including hyperfine
splittings, in the following section. Following which, we discuss our estimates for various
charm baryons, and their comparison with our previous calculations and other lattice
determinations in literature.
5.4.1 Hyperfine splitting in 1S charmonia
The hyperfine splitting in 1S charmonia is one of the most well studied physical quan-
tities in lattice charmonium calculations over the year, and until very recently [72, 73]
Spectroscopy using chiral fermions 110
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
mπa
(M
eV
)
ma
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
mπa
(M
eV
)
ma
Figure 5.4: mπ versus mq for a broad range of quark masses, where the computationswere performed, for (left) coarser and (right) finer lattices. Our quark masses ’ma’ arealways less than 1 and mca = 0.29 and 0.425 for finer and coarser lattices respectively.
100
105
110
115
120
125
130
135
140
0 10 20 30 40 50 60 70
HF
Scc
(vec
tor
- psc
ala
r)
time
a = 0.0582 fm
Figure 5.5: Effective hyperfine splitting in 1S charmonia for wall-point correlators forlattices with spacing 0.0582 fm. Horizontal lines show the fit results with one sigmaerror.
lattice results were found to be smaller than the experimental value (∼ 116 MeV). This
underestimation is now understood to be mainly due to the discretization error associ-
ated with the charm quark action and the quenched approximation. The effects due to
the discretization errors are evident in the results from the excited state spectroscopy
calculations using a tree level tadpole improved cs value, where this hyperfine splitting
is found to be ∼40MeV below the experimental value. In this study, we calculated this
splitting. In Figure 5.5, we show the effective splittings between vector and pseudoscalar
Spectroscopy using chiral fermions 111
correlators (jackknifed) at the tuned charm mass for wall-point correlators on finer lat-
tices. horizontal lines shown are the fit results, with one sigma error bar. Our final
estimate for this hyperfine splitting are 125(6) MeV and 119(3) MeV corresponding to
coarser and finer lattices respectively. A com
0
100
200
300
400
500
600
-0.02 0 0.02 0.04 0.06 0.08 0.1
Mas
s S
pli
ttin
g (
MeV
)
Lattice Spacing (fermi)
J/Ψ − ηc
χc0 − ηc
χc1 − ηc
hc − ηc
Ds* − Ds
Ds0 − Ds
Ds1 − DsDs − ηc/2
Figure 5.6: Meson mass splitting for charmonia and charmed-strange mesons at twolattice spacings. Experimental values are shown in the left side.
5.4.2 Energy splittings in charmonia and charmed-strange mesons
Beside 1S hyperfine splittings, it is also important to consider energy splittings between
various other charmonia. In Figure 5.6, we plot energy splittings between axial, scalar and
tensor charmonia from pseudoscalar charmonium. In addition to this, we also calculate
charmed-strange mesons with various quantum numbers. Energy splittings between these
mesons are plotted in Figure 5.6. It was observed that tuning of charm mass by using
kinetic mass has brought these splittings more closer to experimental values than those
obtained using the rest mass [127]. After adding one more lattice spacing, the continuum
extrapolation will be carried out in future.
5.4.3 Charmed baryons
Our main motivation is to compare baryon masses calculated in this formalism with those
obtained from the excited state spectroscopy study. Here we investigate that by studying
the ground state spectra of charm baryons with one or more charm quark content, e.g.,
Spectroscopy using chiral fermions 112
with quark content css, ccs, and ccc on these lattices. To project out the baryon states
with 1/2 spin, we use the following standard local interpolating operators :
Ωcc ⇒ εabc[cTaCγ5sb]cc and Ωc ⇒ εabc[sTaCγ5cb]sc,
with C = γ4γ2 being the charge conjugation matrix. For spin 3/2 baryons, we choose the
following operators :
Ωccc ⇒ εabc[cTaCγµcb]cc, Ωcc ⇒ εabc[cTaCγµsb]cc and Ωc ⇒ εabc[sTaCγµcb]sc.
However, these operators has both spin 1/2 and 3/2 projections in them. At zero mo-
mentum the corresponding correlation functions using these operators can be written
as
Cij(t) = (δij −1
3γiγj)C3/2(t) +
1
3γiγjC1/2(t),
where i and j are the spatial Lorentz indices and C3/2(t) is the correlation function for
spin 3/2 state [179]. By choosing the appropriate Lorentz components, the spin 3/2 part
of the correlation function, C3/2(t), can be extracted and used to calculate the mass of
the spin 3/2 baryons.
The discretization effects will be maximum for triply charmed baryons since they have
three heavy charm quarks, followed by doubly charmed baryons made with two charm
quarks. Hence we first compare the results of Ωccc obtained from these two independent
studies. The mass splittings Ωccc − 32J/Ψ are shown in Figure 5.7. The factor 3/2
accounts for the difference in the valence charm quark content in the baryons and the
mesons. The blue rectangles represent the results from this work for the two different
lattice spacings, while the rectangles in red are results from the work discussed in the
previous section. The two red points corresponds to the estimates from the tree-level
clover coefficient cs = 1.35 and at a boosted value of cs = 2 (see Section 4.1.2), which
gives correct charmonium hyperfine splitting, J/ψ−ηc. The consistency between the red
and blue squares is evident. We also plot results from other lattice calculations (violet
: Ref. [87], green : [88]). These can also be seen to be consistent with our estimates.
We have omitted the estimate from Ref. [84], as the error bar was very large, though
consistent with our estimates. Different variety of discretization used in above references.
Our calculations were with anisotropic clover and overlap fermions, while Ref. [87] used
Brillouin quraks and Ref. [88] used relativistic heavy quark action for the heavy quarks.
The agreement between our calculations and furthermore with results from other lat-
Spectroscopy using chiral fermions 113
HSC-13 ILGTI-2013 BMW-2012 PACS-CS-2012 RTQM RQM BAG0.06
0.09
0.12
0.15
0.18
0.21
0.24m
ass
Ωccc -
3/2
mass
J/ψ (
GeV
)
cSW
=1.35
cSW
=2.0a=0.0582fm
a=0.09fm
a=0.0888fma=0.07fm
Figure 5.7: Mass splitting of the ground state of JP = 32
+Ωccc from J/ψ meson. A
factor 3/2 is multiplied with J/ψ mass to account for the difference in the number ofcharm quarks in baryons and mesons. This mass splitting mimics the binding energyof the ground state Ωccc. Results of this mass splitting from this work (blue boxes) arecompared with those obtained from calculations discussed in the previous section (redbox). Along with we compare estimates that are available from various lattice and quarkmodel calculations available in the literature.
tice calculations gives us confidence in our estimates from the excited state spectroscopy
calculations. While this consistency in results also make our calculations using overlap
fermions itself serve an excellent formalism to study heavy quark systems. We also quote
a few quark model calculations (orange) in the Figure 5.7, to show how potential mod-
els stand in predictions for the triply charm systems. As one can see that, though the
relativistic calculations by Martynenko [50] and bag model calculations by P. Hasenfratz
[30] lie in the range of lattice predictions, a relativistic calculation by Migura, et al.,
accounting for the spin dependence of the confining potential in two different ways gives
widely different estimates [51]. This plot can potentially provide crucial input to the
future discovery of Ωccc.
Next we calculate various spin splittings for doubly charm baryons where one also
expect larger discretization error. We have already shown the numbers from this calcu-
lation in Figure 4.18 and Figure 4.24 while comparing the numbers from the previous
Spectroscopy using chiral fermions 114
500
550
600
650
700
750
Ωcc −
J/ψ
(M
eV
)
1/2+
3/2+
ILGTI
JP
HW
BALI
HSC
0
40
80
120
hfs
Ωcc
(MeV
)
Figure 5.8: Mass splittings of Ωcc − J/Ψ. The inset plot shows the hyperfine splitting inthis channel. Also shown are other lattice determinations.
calculations with results from various lattice calculation. For a completeness, we show
the results here again. We find our results from the excited stated spectroscopy calcula-
tions and the calculations using overlap fermions are consistent with each other. We also
find similar consistency of our results with estimates from other lattice calculations. In
Figure 5.8, we show the results for 1/2+ and 3/2+ ground states for the Ωcc baryon in
terms its splitting from mJ/ψ. Along side we plot estimates from other lattice determi-
nations. It is to be noted that these baryons are yet to be measured experimentally. The
inset plot shows the corresponding hyperfine splitting. Our results for both the splittings
are consistent with other lattice results [80, 84, 86, 88, 114]. This agreement further
boosts the confidence in our results from our excited stated spectroscopy calculation and
the applicability of overlap fermions in heavy quark systems. In Figure 5.9, we plot our
estimates for the Ωc baryons. It shows the 1/2 and 3/2 ground states for the Ωc baryon
for both the parities. We also quote the experimental numbers also in the plot, so as to
have a comparison. While there are observations for the positive parity states, the neg-
ative parity states are yet to be measured experimentally. In the inset plot, we show the
hyperfine splitting in this baryon, and make a comparison with other lattice estimates.
Once again the consistency in our findings validates the credibility in our results.
Spectroscopy using chiral fermions 115
2600
2700
2800
2900
3000
3100
3200
3300
Ωc (
MeV
)
1/2+
3/2+
1/2-
3/2-
0
40
80
120
hfs
Ωc (M
eV
)
ExptILGTI
JPHW
BALIHSC
MATHUR
Figure 5.9: Ground states of Ωc for even and odd parities. The inset plot shows thehyperfine splitting in this channel. Also shown are other lattice determinations, alongwith the experimental values, where available.
5.5 Summary, conclusions and future prospects
In this chapter, we discuss an alternative dynamical lattice QCD calculation of the low
lying heavy hadron states. The main motivation is to compare the results between
two independent calculations by two different type of discretization and then get an
estimate of the discretization effects in energy spectra. Using a mixed action approach,
we utilized the highly desirable features of small discretization errors and small taste
breaking effects of the HISQ action and the small discretization effects and the low quark
mass limit of the overlap fermion action. We are thus able to study a long range of
pseudoscalar meson masses from 230 − 5000 MeV. By tuning the charm quark mass
using the kinetic mass, we made sure the discretization errors due to the uncertainty in
the charm mass tuning is small. And at this tuned charm mass, we extracted the meson
and the baryon spectra containing one or more charm quarks. We observed that results
from our two independent methods are consistent with each other. Furthermore, we made
a comparative study of our results with results from other lattice calculations and quark
model calculations. The consistency within the lattice calculations and the agreement of
those with the experimental values, wherever available, validates the credibility in both of
our calculations. Thus, this calculation serves simultaneously as a complementary study
to compare the results from our previous calculations and as an independent study with
improved control over the systematics that gives predictions about the low lying charm
hadron spectrum.
Chapter 6
Baryons at finite temperature
6.1 Introduction
In this chapter, we will describe our investigations of the nature of the strongly interacting
matter at high temperatures using baryonic probes. The QCD matter undergoes a change
of phase when its temperature is varied: from a chiral symmetry broken state of color
singlet objects called hadrons at low temperatures to a deconfined chiraly symmetric state
of the quark gluon plasma at high temperatures. It has been observed from many lattice
calculations that there is no sharp phase transition but a cross-over from the hadronic to
the deconfined state. One the other hand, in a pure gluonic field theory, this change of
phase proceeds through a first order transition. In this work, we focused on the hadronic
correlation lengths that exist in a pure SU(3) gauge theory above and immediately below
the deconfinement transition temperature, Tc. These correlations will help us understand
the non-perturbative nature of QCD with varying temperature.
The starting point of the lattice computations of the correlation lengths in QCD
medium at finite temperature, T, is the thermodynamic partition function,
Z = Tre−βT HQCD, (6.1)
where HQCD is the QCD Hamiltonian and βT = 1/T (kB = 1). This partition function
can be expressed as a path integral of the QCD action, SQCD (eq. (3.5)), over a four
dimensional Euclidean space-time with a compactified temporal direction of extent βT =
1/T ,
ZT =
∫DψDψDAµe
−SQCD . (6.2)
116
Baryons at finite temperature 117
At finite temperature, the fermion and the gauge fields must satisfy anti-periodic and
periodic boundary conditions along the temporal direction respectively due to the trace
relation
ψ(x, t+ 1/T ) = −ψ(x, t), Aaµ(x, t+ 1/T ) = Aaµ(x, t) (6.3)
After discretizing the space-time, the physical volume of the system is V = N3s a
3 and
T = 1/(Nta), where Ns an Nt are the number of lattice points along the spatial and the
temporal directions.
We study the QCD medium at finite temperatures by looking at the static correlation
functions,
O(z) =1
ZTr(h(z)h(0)e−βT HQCD) and
O(z) |z→∞ = b exp [−µ(T )z]. (6.4)
Static correlation lengths (µ) indicates the spatial distance beyond which the effects of
putting a test hadron in the medium is screened. This response depends on the quantum
numbers carried by the probes; so one can classify static correlators as glueball-like,
meson-like and baryon-like probes with the usual quantum numbers of these quantities
corrected for the fact that the static spatial symmetries are different from the Poincare
group.
Meson-like screening masses have been studied in QCD in great detail [128, 129, 130,
131, 132, 133, 134, 135]. Baryons at finite temperature has not been studied in detail
in recent past and a few notable works were in the late 1980’s [128, 133]. Moreover, a
detailed study of the baryon screening masses in the low temperature phase also has not
been performed yet. In this work, we perform simulations of pure gauge theory for three
different temperatures across the transition temperature and study screening correlators
for mesonic and nucleonic resonances with clover fermions. The main emphasis, in this
work, has been given to the nucleon channels, which are expected to provide important
inputs to the study of baryon number fluctuations and thus to the experimental search
for the critical point of QCD.
The organization of this chapter is as follows. In Section 6.2 we discuss technical
details related to the simulation, inversion and analysis, including the details of inter-
polating operators used and their asymptotic fit forms. In Section 6.3 and Section 6.4
we display the results for mesonic and baryonic channels, respectively, for temperatures
above and immediately below Tc. In Section 6.5, we compare our results on screening
Baryons at finite temperature 118
masses with the estimates from free theory and the other existing lattice calculations.
Section 6.6 summarizes and concludes this study of hadrons at finite temperature. Ap-
pendices 6.A and 6.B discusses two technical aspects related with these calculations,
which interested reader is referred to in the main section in this chapter.
6.2 Runs and measurements
T/Tc Nτ ×N3s β Nconf cSW κ
0 32× 163 6.03 71 1.7333 0.1345, 0.13470.95 8× 323 6.03 94 1.7333 0.1345, 0.13471.5 8× 323 6.332 67 1.5667 0.1345, 0.1350, 0.1355
Table 6.1: The simulation and measurement parameters used in this work. β is thebare coupling for the Wilson gauge action, cSW is the clover coefficient (determined non-perturbatively using eq. (2.22)), and κ is the hopping parameter in the improved Wilson-Dirac operator.
The hadronic correlation functions with mesonic and baryonic quantum numbers were
constructed out of clover-improved Wilson quarks [143], in zero and finite temperature
ensembles generated using the Wilson gauge action. The finite temperature configura-
tions were generated on lattices with lattice spacing a = 1/(8T ) with T = 0.95Tc and
1.5Tc. The spatial size of the lattice, L, was tuned such that LT = 4. We use the notation
L = Ns a and V4 = V/T . Zero temperature measurements were carried out on (2L)×L3
lattices with coupling β = 6.03, which corresponds to the same lattice spacing as the
finite temperature run below Tc [180, 181]. The box size, L, is sufficiently large (> 4
times the Compton wavelength of the lightest state on the lattice) in all our lattices, as
a result of which finite volume effects are under control. A zero temperature study at
β = 6.332, corresponding to the thermal ensemble above Tc was not made, but the corre-
sponding choice of κ was made so that we could use a previous study [182]. Details of the
simulation and measurement parameters are tabulated in Table 6.1. With the choices of
κ listed for β = 6.332, the values of mπ/Tc are 1.52, 2.58, and 3.31 respectively.1.
The zero momentum correlation functions for the mesonic channels are,
SH(t) =∑x
〈H†(x, t)H(0, 0)〉, where H(x, t) = ψ(x, t)ΓHψ(x, t), (6.5)
1Discussion on a subtle aspect related to the studies has been omitted from the main text and hasbeen discussed in detail in the Appendix 6.A
Baryons at finite temperature 119
where ψ(x, t) is the quark field at time t and spatial point x (we will also use the com-
ponent notation x1 = x, x2 = y and x3 = z). The sum over all spatial sites of the
point-to-point correlator projects on to zero spatial momentum. All Dirac, flavor, and
color indices are summed. ΓH is an appropriate Dirac, flavor matrix which gives the
quantum numbers of the meson, H. In actual practice, we replaced the point source of
eq. (6.5) by a wall source (see eq. (5.5)). Wall source allows the suppression of high mo-
menta modes, especially above Tc, allowing more accurate study of the screening masses,
which are related to the long distance behavior of the static correlation function in eq.
(6.5). Wall source being a gauge non-invariant source, this required gauge fixing; we fixed
to the Coulomb gauge.
We used ΓPS = γ5 for the isovector pseudoscalar (PS), ΓS = 1 for the isovector scalar
(S), ΓV = γi for the vector (V) and ΓAV = γiγ5 for the isovector axial vector (AV)
channels. For the T = 0 measurements, we considered propagation in the time direction
as shown in eq. (6.5), and since all the orthogonal directions are equivalent, we summed
the V and AV propagators over the three polarizations i = 1, 2, 3. For the screening
correlators all the polarizations of the V and AV are not equal, and we summed over only
i = 1 and 2.
Masses and screening masses can be obtained without renormalization. They were
estimated, as described in Section 3.1, by fitting to an assumed cosine-hyperbolic form,
and looking for agreement with effective masses. Statistical errors on masses are obtained
by a jackknife procedure. The propagation of statistical errors is done by jackknife when
they could be correlated, and by adding in quadrature when they are independent.
(a) (b) (c)
Figure 6.1: Quark line diagrams which couples with scalar propagator in the full theory.In the quenched theory (c) does not contribute.
Though analysis of meson correlators is straight forward, scalar correlators in quenched
calculations require special care. There have been several calculations to study the isovec-
tor scalar meson (a0) at zero temperature with the ψψ interpolation field in the quenched
approximation [183, 184, 185]. In all these works the scalar correlation functions were
found to change the signature between the source point and Nτ/2 violating spectral pos-
itivity. In this work, we make the observation of similar interesting unphysical behaviors
of the scalar correlators at finite temperature (see for example Figure 6.2). An expla-
Baryons at finite temperature 120
nation of this behavior based on quenched chiral perturbation theory (QχPT) has been
discussed in [183, 185]. Unlike full QCD, the contributions to the scalar correlator in
the quenched approximation comes only from diagrams Figure 6.1(a) and Figure 6.1(b).
The diagram Figure 6.1(b) is equivalent to the propagation of η − π intermediate state
along significant fraction of the time in the correlation function, in contrast with the
diagram Figure 6.1(a), which corresponds to the propagation of a scalar excitation. The
negative value of the ψψ correlator appears due to the absence of the diagrams like in
Figure 6.1(c), and hence does not compensate for the coupling of scalar with the η − πghost state appearing due to the presence of Figure 6.1(b). In quenched calculations the
intermediate η − π state is equivalent to π − π state and hence we use twice the mass of
the pseudoscalar excitation in our studies to estimate its binding energy.
-4
-3
-2
-1
0
1
2
3
106*4
4 6 8 10 12 14
CS(z
,t)/
Tc3
7.6*(z,t)*Tc
T/Tc = 0.950
Figure 6.2: Scalar correlation functions below the deconfinement transition temperature,Tc: for temperatures T = 0 (red) and T/Tc = 0.95 (blue) and mπ/Tc = 1.99. The con-tinuous curves are the fit estimates using the asymptotic fit forms predicted by quenchedchiral perturbation theory (eq. (6.6)).
In references [183, 185], it has been argued that the large time behavior of the zero
temperature scalar correlator in the quenched theory is given by
SQχPT (t) = b exp [−mSt] + exp [−mS(Nt − t)]
−aS (1 +mπt) exp(−Eη′πt) + (1 +mπ(Nt − t)) exp(−Eη′π(Nt − t)) . (6.6)
Baryons at finite temperature 121
The negative term is explicitly due to the quenched ghost. mπ/S are the masses of the
PS and S meson, Eη′π is the energy of the ghost state (Eη′π = 2mπ−Eint) and −aS is the
coupling to the ghost. Since the ghost exactly cancels a physical term, these two param-
eters are physical. For sufficiently small mπ the second term dominates at intermediate
distances, leading to negative values for the correlator. Though the asymptotic form in
eq. (6.6) has been derived for zero temperature, in our calculations we have used the
same form at finite temperature to study the thermal effects for the scalar meson. The
continuous lines in Figure 6.2 are the best fit curves describing the unphysical coupling,
obtained by fitting the tail of the correlation functions with the coupling term as in eq.
(6.6). This negative dip becomes prominent as one decreases the quark mass.
The SUL(2)×SUR(2) chiral symmetry of the QCD vacuum is restored above Tc. The
anomalous UA(1) remains broken asymptotically, although its effects are seen to be small
at temperatures of 1.5Tc. The most straightforward consequence of this is that the S/PS
and the V/AV correlators should become pairwise degenerate in the high temperature
phase. The observation of some breaking of this symmetry has been an issue of some
interest recently [134]. Here we introduce a new measure for this symmetry—
RH =1
Nt − 1
Nt−1∑t=1
〈SH+(t)− SH−(t)〉〈SH+(t) + SH−(t)〉
, (6.7)
where SH±(t) are parity partner correlators (for the screening correlators t is replaced by
z and Nt by Ns), and the angular brackets are averages over the gauge ensemble. The
correlators at t = 0 are left out of the sum to avoid problems with time doublers. We
will use the convention of taking the positive parity partner for the label H. When chiral
symmetry is broken we expect RH ' O(1).
One subtlety in computing RH is that the lattice mesonic currents need to be renor-
malized to connect them with the continuum currents, before SH could be used to com-
pute RH. For quarks of mass mq, the lattice operators, H(x, t), are multiplicatively
renormalized to
ZH(g2)(
1 + bH(g2)amq
)H(x, t), (6.8)
where a is the lattice spacing, g2 = 6/β, and amq = (κ − κc)/2. We use the results
in the MS scheme where the renormalization constants are determined at the scale of
1/a. For non-perturbatively improved clover fermions, the factors ZV ,AV and bV have
been calculated non-perturbatively [186], where the following interpolating formulae are
Baryons at finite temperature 122
found:
ZV =1− 0.7663g2 + 0.0488g4
1− 0.6369g2, ZAV =
1− 0.8496g2 + 0.0610g4
1− 0.7332g2,
bV =1− 0.6518g2 + 0.1226g4
1− 0.8467g2. (6.9)
For the other coefficients, we use the expression from the one-loop, tadpole improved
perturbation theory [187]:
ZPS,S = u0
(1− zPS,Sg2
), (6.10)
where u0 is the tadpole factor, 〈P 〉 = u40, g2 = g2/u4
0, and the one-loop coefficients zPS,S
as obtained from [188] are zPS = 0.107−0.019cSW +0.017c2SW and zS = 0.026+0.065cSW−
0.012c2SW where cSW = u3
0cSW . The one-loop order tadpole-improved forms of bH can be
written as
bH =1
u0
(1 + b1
H g2). (6.11)
where one obtains from the calculations of [189] the values b1PS = 0.109, b1
S = 0.070, and
b1AV = 0.069, computed for the tree-level cSW = 1.
For the nucleon operator we used
Nα(x, t) = εabc(Cγ5)βδψaα(x, t)ψbβ(x, t)ψcδ(x, t), (6.12)
where α, β, and γ are Dirac indices, a, b, and c are color indices, ε is the Levi-Civita sym-
bol, and C is the charge conjugation operator. The projection of the correlation function
on to vanishing spatial momentum is performed by the usual means of summing over x
at T = 0 where we imposed periodic boundary conditions in all directions. However, at
finite temperature, anti-periodic boundary conditions must be imposed on quark fields in
the Euclidean time direction. As a result, the screening correlator must be projected to
the lowest Matsubara frequency [128]. The parity projection operators for the correlator
whose propagation is measured in the direction µ is
Pµ± =1
2(1± γµ). (6.13)
The zero momentum correlators of the two parities of the nucleon can be fitted to the
Baryons at finite temperature 123
behavior expected of a Wilson fermion—
SN+(t) = c+ exp [−µN+
t] + c− exp [−µN−(Nt − t)]
SN−(t) = c− exp [−µN−t] + c+ exp [−µN+(Nt − t)]. (6.14)
Because of the admixture from opposite parity states, these correlators change sign across
the middle of the lattice. It is to be noted that this structure is also expected for the
excited baryon spectrum for both the parities. It is because our construction and the
analysis were always concentrated on the temporal ranges close to the source point in the
forward temporal direction, we did not encounter this behaviour in Chapter 4. In this
work, we study this structural behaviour in the nucleon correlations to study the chiral
symmetry across Tc. For this, we used the absolute value of the correlators above. We
note that RN does not require knowledge of ZN , because the parity partners are generated
by the same lattice operator. For the T = 0 measurement, where µN± = mN± . For the
finite temperature measurement where we replace t by z, we have
µ2N± = µ2
N± − sin2(π/Nt), (6.15)
where we have subtracted the contribution coming from the anti-periodic boundary along
the temporal direction [128].
6.3 The meson sector
κ mπ/Tc mV /Tc mAV /Tc0.1345 2.20± 0.03 3.55± 0.06 5.6± 0.40.1347 1.99± 0.03 3.53± 0.08 5.5± 0.5
Table 6.2: Meson masses in units of Tc at T = 0 and β = 6.03.
The analysis of meson masses at T = 0 follows the procedure described in Section 3.1.
The results are collected in Table 6.2. We choose to express all results in units of Tc for
two reasons. First, because quenched QCD is not open to experimental tests and hence
quoting numbers in MeV units is based on assumptions which cannot be tested. We
prefer to quote ratios of quantities, which are computable in practice. Second, because in
quenched QCD the critical coupling is known with high precision, and Nt = 8 is within
Baryons at finite temperature 124
the scaling region [190], the statistical and systematic errors involved in using this as a
scale are completely under control.
mπ/Tc H µH/Tc µH/mH
2.20 PS 2.18± 0.02 0.99± 0.02V 3.45± 0.06 0.97± 0.02
AV 5.18± 0.09 0.93± 0.061.99 PS 1.97± 0.02 0.99± 0.02
V 3.31± 0.07 0.94± 0.03AV 5.1± 0.1 0.92± 0.08
Table 6.3: Meson screening masses, µH at T = 0.95Tc in units of Tc and the correspondingT = 0 meson mass, mH.
The analysis of most mesonic correlators below Tc is equally straightforward. The
only subtlety has been mentioned earlier: since the zero momentum screening correlator is
measured for separations along the z-direction, the three polarizations states of the V and
AV are two spatial and one temporal. The temporal polarizations have behavior distinct
from the spatial polarizations [191]. We measure the screening masses of the spatial
polarizations only. Our main results for the screening masses below Tc are summarized
in Table 6.3. Our results shown in Table 6.3 indicates that the pole mass of the mesons
is hardly affected by temperature.
T = 0 T = 0.95Tcmπ/Tc Eint/mπ χ2/dof Eint/mπ χ2/dof2.20 0.6± 0.2 1.92 −0.1± 0.8 0.251.99 0.5± 0.2 1.28 0.1± 0.7 0.57
Table 6.4: Eint in units of mπ obtained by fitting the scalar correlator with the fit formin eq. (6.6).
We treat the scalar correlator separately because of its distinctive behavior in corre-
lation functions compared to those for other mesons as shown in Figure 6.2. Table 6.4
contains the measure of the binding energy (Eint), in units of mπ, extracted by fitting
the scalar correlation functions with the functional form in eq. (6.6) at both T = 0 and
for T < Tc. Eint at T = 0 is non-zero at the 95% significance level for both values of
mπ which we used. While the finite temperature measurements are statistically indis-
tinguishable from these, they are consistent with zero within errors, due to the larger
Baryons at finite temperature 125
fit errors at T > 0. It is interesting to note that the central values of Eint at T > 0 lie
outside the 2σ errors of the T = 0 measurement.
102
103
104
105
106
107
108
0.5 1 1.5 2 2.5 3 3.5
SP
S,S
(z)/
T3
zT
PSS
mπ/Tc= 1.52
(a)
102
103
104
105
106
107
108
0.5 1 1.5 2 2.5 3 3.5
SV
,AV(z
)/T
3
zT
VAV
mπ/Tc= 1.52
(b)
Figure 6.3: A signal of restored chiral symmetry above Tc is that (a) the PS and Scorrelators and (b) the V and AV correlators become degenerate. The continuous curvesare correlation functions in a FFT; in order to remove trivial artifacts, these have beencomputed on a lattice of the same size.
As described in the previous section, chiral symmetry restoration in the high temper-
ature phase of QCD is signaled by pairwise equality of correlators which are related to
each other by parity. This is the case shown in Figure 6.32. The figure also shows that the
correlation functions are not too far from those expected in a theory of non-interacting
quarks (also called the free field theory, FFT).
At T = 0 in the V/AV sector we found RV = 0.73 ± 0.02; at T = 0.95Tc this drops
marginally to RV = 0.59 ± 0.02 for both the bare quark masses we have used. This
indicates that chiral symmetry remains strongly broken up to 0.95Tc. A simple model of
V/AV mixing below Tc was presented in [192] using a mixing parameter ε, which can be
adapted to our use by writing schematically
SV (T ) = (1− ε)SV (0) + εSAV (0), and SAV (T ) = (1− ε)SAV (0) + εSV (0). (6.16)
This gives RV (T ) = (1−2ε)RV (0). Using the values quoted above, we find ε = 0.10±0.02.
In the unquenched theory one may expect much larger values of this parameter below
Tc [192]. We observed that at T = 1.5Tc the value of RV drops as a power of the quark
mass, vanishing as mπ vanishes (see Figure 6.4). However, there is no dependence of RS
2The quantity in the x-axis is a dimensionless measure of the lattice extension in units of the tem-perature, T . It is thus labeled, and runs from 0→ 4, because for the finite temperature lattices LT = 4.
Baryons at finite temperature 126
on the quark mass.
10-4
10-3
0.01
0.1
1 2 3 4
RH
m /Tcπ
V
S
Figure 6.4: Above Tc, RS is found to be almost independent of mπ/Tc. However, RV goesto zero as a power of mπ/Tc, showing that correlation functions of the parity partners Vand AV do become exactly degenerate in the limit.
We have extracted screening masses, µH from the correlation functions. We found that
effective masses showed a good plateau which agreed with fits, unlike previous experience
with Wilson quarks at T > Tc [130]. The main reason for this is the use of wall source and
the details are explored in Appendix 6.B instead of a point source, where some effective
mass plateaus are also displayed. Our results for the screening masses are collected in
Table 6.5. The pairwise degeneracy of the masses is quite evident. It is also evident that
the screening masses are close to that expected in a theory of free quarks. The screening
masses in the S/PS channel are 7–9% smaller than that in the free theory, whereas in
the V/AV channel they are within 2–3% of the free theory.
mπ/Tc µPS/T µS/T µV /T µAV /T µN+/T µN−/T
1.52 5.54± 0.02 5.59± 0.04 5.88± 0.02 5.89± 0.02 8.72± 0.10 8.68± 0.102.58 5.56± 0.02 5.59± 0.02 5.90± 0.02 5.91± 0.02 8.75± 0.10 8.73± 0.103.31 5.61± 0.02 5.65± 0.02 5.95± 0.02 5.96± 0.02 8.82± 0.08 8.77± 0.09
Table 6.5: Hadron screening masses at T = 1.5Tc for three different masses of quarks.In FFT all the mesonic (baryonic) screening masses are expected to be 5.99 (8.44),5.995 (8.456) and 6.01 (8.48) on our lattices, when the bare quark mass is tuned to givemπ/Tc = 1.52, 2.58 and 3.31 respectively.
Baryons at finite temperature 127
6.4 The baryon sector
103
104
105
106
107
108
109
0.5 1 1.5 2 2.5 3 3.5 4
SN
+,-(t
)/T
c3
t Tc
mπ/Tc = 1.99
N+N-
(a)
101
102
103
104
105
106
107
108
0.5 1 1.5 2 2.5 3 3.5
SN
+,-(z
)/T
c3
zT
mπ/Tc = 1.99
N+N-
(b)
Figure 6.5: Nucleon correlators at (a) T = 0 and (b) T = 0.95Tc. Both sets are asym-metric, showing that the nucleon, N+ and its parity partner, N−, are not degenerate.
Analysis of baryon correlators requires more care than meson correlators. This is be-
cause states of opposite parity contribute to each parity projected correlator, as explained
in eq. (6.14) and shown in Figure 6.5. This contribution from the parity partners that
are non-degenerate due to chiral symmetry breaking causes the asymmetric behavior in
the nucleon correlation function, as is evident from Figure 6.5(a). As a result, both the
fitting procedure and the extraction of effective masses is more complex than the analysis
for mesons. In addition, the projection of the screening correlator on to zero momentum
requires a twist to compensate for the thermal boundary condition [128]. The subsequent
extraction of a screening mass from the nucleon correlator also requires the subtraction
of (see eq. (6.15)) resulting in additional loss of precision.
In Figure 6.5, it would appear that the finite temperature correlators are more nearly
symmetric than those at T = 0. Whether or not there is an early onset of chiral symmetry
restoration can be probed by constructing the measure RN , defined by eq. (6.7). At
T = 0 we find RN = 0.88 when mπ/Tc = 2.20 and 0.89 at the lower quark mass. At
finite temperature these change to RN = 0.8 and 0.83 respectively. Contrary to the visual
impression created by Figure 6.5, once the covariances between the N± correlators are
accounted for, the correlation functions themselves do not show any tendency towards
early restoration of chiral symmetry. In fact if we adapt the model of eq. (6.16) to this
case, we find a mixing parameter ε = 0.04 in the nucleon sector, which is even smaller
than that found for the V/AV at the same temperature.
Baryons at finite temperature 128
mπ/Tc mN+/Tc µN+
/Tc mN−/Tc µN−/Tc2.20 5.10± 0.10 5.3± 0.2 6.8± 0.4 6.6± 0.41.99 4.9± 0.1 5.28± 0.14 6.7± 0.5 6.4± 0.7
Table 6.6: Nucleon masses at T = 0 and screening masses at T = 0.95Tc.
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 12
(mN
- - m
N+) e
ff
t
(a)
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 12
(µN- −
µN+) eff
z
(b)
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 12
(mN
- - m
N+) e
ff
t
(c)
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 12
(µN- −
µN+) eff
z
(d)
Figure 6.6: The analysis of mass splittings between N+ and N− for mπ/T = 2.20 at (a)T = 0 and (b) T = 0.95Tc and for mπ/T = 1.99 at (c) T = 0 and (d) T = 0.95Tc. Thedata points are effective masses, the band between the dashed lines is obtained by takingthe difference of screening masses. The band between the full lines is obtained by directfits to the splitting between the two masses, and gives the results shown in eq. (6.17).
The results of fitting the masses is shown in Table 6.6. Here we indeed find some
precursor effects of chiral symmetry restoration in the form of a thermal shift in the mass
splitting between the baryon and its parity partner. We measured the splitting by the
Baryons at finite temperature 129
ratio of the correlators, SN+/SN− , and extract the mass difference, ∆m, by fitting to an
exponential form exp(−∆mt). Since this method takes care of covariances between the
correlators SN+and SN− , we expect to control statistical errors better. The fit can be
cross checked against the equivalent of effective masses for the ratio. Such checks are
exhibited in Figure 6.6. The resulting values are
mN− −mN+
Tc=
1.53± 0.09 (mπ/Tc = 2.20)
1.82± 0.17 (mπ/Tc = 1.99)
and
µN− − µN+
Tc=
1.15± 0.14 (mπ/Tc = 2.20)
1.24± 0.20 (mπ/Tc = 1.99). (6.17)
mπTc
µVmV
µN+
mN+
µAV −µVmAV −mV
µN−−µN+
mN−−mN+
2.20 0.97± 0.02 1.05± 0.05 0.86± 0.16 0.75± 0.101.99 0.94± 0.03 1.07± 0.04 0.89± 0.22 0.68± 0.12
Table 6.7: Thermal shifts in masses and mass splittings at T = 0.95Tc; µH denotes thescreening length in the hadronic channel H, and mH the mass at T = 0. The ratio of masssplittings suggest that, unlike in the case of vector-like mesons, nucleonic correlationsshow some precursor effects to chiral symmetry restoration immediately below Tc.
In Table 6.7, we show the thermal shifts in the masses and the mass splittings in
nucleonic and vector-like parity partners, in terms of the ratio of masses and the ratio
of mass splittings between non-zero temperature and zero temperature. We find that
µH = mH within 95% confidence limits, indicating no significant thermal effects in the
vector meson, V and nucleon, N+. Nor do we observe a thermal shift in the mass
splittings between the V and the AV (axial vector) mesons. However, we find that in the
nucleon sector the mass of the opposite parity nucleon moves closer to the ground state
immediately below Tc. This is the most definite evidence to date about precursor effects
to chiral symmetry restoration immediately below Tc. The thermal effects in the parity
partner nucleon is, hence, significant and these observations not only constrain models
of quantum hadrodynamics [192], but could also have implications for the analysis of
heavy-ion collision data.
Above Tc the correlators for N± become symmetric and degenerate (see Figure 6.7).
We find RN = 0.230 ± 0.005 when mπ/Tc = 3.31, RN = 0.146 ± 0.006 for mπ/Tc = 2.58
Baryons at finite temperature 130
10-2
10-1
100
101
102
103
104
105
106
107
108
0.5 1 1.5 2 2.5 3 3.5
CN
+,-(z
)/T
3
zT
N+N-
Figure 6.7: Nucleon and its parity partner correlators at 1.5 Tc. The degeneracy inthe correlators signal the restored chiral symmetry at this temperature. The continuouscurves are correlation functions in a FFT; in order to remove trivial artifacts, these havebeen computed on a lattice of the same size.
and RN = 0.076± 0.009 for mπ/Tc = 1.52. These lead to a vanishing of RN as a power of
mπ/Tc. Chiral symmetry restoration is also seen in the fitted screening masses, displayed
in Table 6.5. As in the meson sector, the screening masses are within 3–4% of those
expected in a theory of free quarks.
6.5 Hadrons above Tc and free field theory
Above Tc, at the temperature T = 1.5Tc, we found strong signals of approximate chiral
symmetry restoration in the near degeneracy of screening masses of all the hadronic
parity partners, evident in Table 6.5. We also observed the correlation functions and
screening masses are not far from those obtained in FFT, which is a model of non-
interacting quarks. For mesonic channels, this observation has been already made in
Refs. [129, 132, 130, 131, 134]. To make a quantitative statement on these observation, in
the Figure 6.8, we show the ratio of screening masses for mesonic and nucleonic quantum
channels at T = 1.5Tc in the interacting theory to the free theory. Alongside we also
show the estimates from other lattice calculations. We see that the PS and S screening
Baryons at finite temperature 131
0.5
0.6
0.7
0.8
0.9
1
1.1
PS SC V AV N+ N−
µh/µ
hF
FT
clover 2013clover 2006 [130]
overlap 2008 [131]staggered 2002 [132]staggered 1988 [133]
Figure 6.8: The ratio of screening masses measured at T = 1.5Tc in quenched QCD withthose in FFT. Along side we add the estimates for these ratios from earlier computationswith clover [130], overlap [131] and staggered fermions[132, 133]. Unity on the y-axisindicates that the hadron propagates as freely propagating quarks in the absence ofgluonic fields.
masses are only ∼8% away from the free theory estimates, while the V, AV and nucleonic
channels already are consistent with a model of non-interacting quarks. We also observe
that our estimates agree with previous work using clover quarks [130]. It also agrees with
results of studies using overlap quarks at smaller lattice spacing [131]. However, it seems
that studies with naıve staggered quarks always indicate stronger deviations from FFT
than any of these above studies, indicating strong staggered artifacts in the estimates.
Later studies using dynamical staggered fermions, with appropriate smearing to account
for the staggered cut off artifacts, related to the taste-splitting, have shown that this is
indeed the case and the estimates from this study were found to be much closer to the
FFT [193].
6.6 Summary, conclusions and future prospects
In this work, we studied the hadronic screening correlators above and immediately below
the deconfinement transition temperature, Tc, using non-perturbative lattice method in
Baryons at finite temperature 132
the quenched approximation with lattice spacing 1/8T employing the clover improved
Wilson quarks. We perform simulations at temperatures T/Tc = 0, 0.95 and 1.5. As seen
in the other lattice calculations, we observed no statistically significant thermal effects
in the mesonic correlations immediately below Tc. We also make the observation of the
unphysical quenching artifacts in the scalar screening correlators immediately below Tc
and make a first attempt to understand the thermal effects on the ghost coupling in these
correlators. We found no statistically significant deviations with the available statistics in
these studies also. Above Tc, clear evidence of the chiral symmetry restoration in terms
of the pairwise degeneracy in the parity partner correlators and the weakly interacting
nature of quarks was observed in the mesonic channels.
Above Tc, baryonic correlations also show clear evidence of the chiral symmetry
restoration in terms of the pairwise degeneracy in the parity partner correlators and
propagate as three weakly interacting quarks. Below Tc, we find interesting behavior in
the nucleonic correlations, indicating precursor effects for the chiral symmetry restoration
in the nucleonic channels immediately below Tc. While no such effects have been seen
earlier either in the glue sector or with quarks, these observations are expected to have
implications for the analysis of the heavy ion collision data. These implications clearly
call for follow-up studies of nucleon below Tc.
Appendix
6.A Exceptional configurations
In our calculations we observe certain configurations with widely different behavior in
the pseudoscalar correlator in comparison with the pseudoscalar correlators from the
other majority of configurations. The pseudoscalar correlators in these configurations
were found to be very much asymmetric about the center of the lattice with respect to
the source and oscillating with unexpected features, which is different from the expected
cosine hyperbolic behavior. A typical example of pseudoscalar correlator of a few of such
configurations observed in this work is shown in Figure 6.9.
Configurations with such unexpected behavior have been observed in quenched cal-
culations by various collaborations [130, 194, 195, 196, 197, 198] using the same fermion
discretization as in this work. Pseudoscalar charge [196] in these configurations were
found to show singular behavior below the critical hopping parameter value (κc) indicat-
Baryons at finite temperature 133
103
104
105
106
107
108
109
0.5 1 1.5 2 2.5 3 3.5
Cps(t
)/T
3
zT
allbadbadbad
Figure 6.9: A typical case of appearance of exceptional configurations in pseudoscalarcorrelators. The filled circles are identified as correlators in exceptional configurationsbased on our analysis. These correlators are at T = 1.5Tc at mps = 250 MeV .
ing a shift in κc value corresponding to these configurations [195, 196]. In literature these
configurations are called as exceptional configurations. An explanation for appearance
of such configurations is as follows. The real part of all the eigenvalues of a discretized
fermion operator (eg. clover improved Wilson fermions) with explicit chiral symmetry
breaking term is not bound to have same signature. It can happen that a subset in the
sampled set of configurations possess an eigenvalue near to −m and hence D +m is not
invertible. In the dynamical simulations, these configurations will not be contributing as
the fermion determinant will be zero. But in quenched calculations the non-zero measure
of configurations having such unphysical zero modes causes unexpected behavior in the
correlation functions and limits the quark mass to values much heavier than the physical
quark mass, that can be studied using such operators[194].
There were attempts to ameliorate this problem in different ways in literature. AL-
PHA collaboration [199] used a twisted mass term in the lattice fermion operator to
remove the singularities in the quark propagator due to the exceptional configurations.
It was, however, shown that the twisted mass term doesn’t solve the problem of excep-
tional configurations in the deconfined phase [200]. In the light hadron spectroscopic
studies, [195] used fat-link improved clover fermions successfully to improve the chiral
Baryons at finite temperature 134
properties of the operator and to suppress the unphysical configurations. However it was
observed that the attractive short distance piece of the potential gets modified by the
fattening. This results in serious systematic errors in the decay constant and quarkonium
spectroscopic studies [201, 202]. Since one of the main objects of our study is the cou-
pling of the scalar with the η′ − π ghost state in the finite temperature scalar screening
correlators, we haven’t used this technique to get rid of these unphysical configurations.
Ref. [196] employs a modified quenched approximation in their work, in which they
push the shifted κc in such configurations back to the average value of κc, which are
determined from the chiral limit of the pseudoscalar masses in the well-behaved config-
urations. The method lacks a systematic way of labeling a configuration as exceptional,
because even two ’well-behaved’ configurations needn’t have same κc always. Moreover,
the overhead calculations needed to determine the zero mode in such configurations is
also significantly large. Ref. [130] reported the observation of configurations with such
unexpected behavior and to get rid of these issues, they handpicked and discarded such
configurations during the analysis. In the present work, we follow a similar approach, but
a systematic procedure developed based on phenomenology of such unphysical configura-
tions. In comparison with most of the other techniques, our analysis requires very little
overhead computation. Also this method could be supplemented in either of the methods
in [196] or [130] to classify such configurations. Here we discuss the naive phenomeno-
logical approach we have employed to exclude the configurations with such unexpected
behavior.
We used the pseudoscalar correlators to define two measures, the measure of symmetry(S)
and the measure of oscillation(O). We defined cuts in these measures such that varying
the measures slightly around the cut will not cause significant changes in the estimates.
Those configurations which lie outside the defined cuts were excluded from the analysis.
Below we discuss the details of the measures we defined and the details of the analysis
done.
• Measure of symmetry (S(i))
We expect the pseudoscalar correlator to be well-behaved and symmetric about
the center in a periodic lattice. S, a measure of the symmetry in the pseudoscalar
correlator, is defined as in eq. (6.18). The superscripts (i) are the configurations
labels.
Baryons at finite temperature 135
S(i) =
Nτ/2+1∑t=2
α(i)t
2(6.18)
where α(i)t =
C(i)(t)− C(i)(Nτ + 2− t)C(t)
and C(t) =1
2N
N∑i=1
(C(i)(t) + C(i)(Nτ + 2− t))
(a) (b)
Figure 6.10: Histograms of the quantity (a) S and (b) O. All the configurations on lefthand side of the blue dotted line are considered as well-behaved configurations.
• Measure of oscillation (O(i))
The pseudoscalar being the lightest, oscillations in the correlator is very unlikely
and can occur only to the level of statistical fluctuations. We introduce a quantity,
O(i), which is a measure of the oscillations observed in a correlator. It is defined as
follows. Those configurations with the minima of the correlator lying outside the
range > ±3 time slices from the center of the lattice with respect to the source point,
are considered ill-behaved. Such configurations are removed before estimating the
measure O(i). In each of the remaining configurations we count the no. of points
a(b) where the slope is positive(negative) in the first(second) half of the correlator
excluding the middle band defined earlier. Then, O(i) is defined to be a+ b.
The histograms of the measures for the lightest pseudoscalar mass at zero temperature
and T = 0.95Tc are shown in Figure 6.10. The blue dotted line is the cut used in the
Baryons at finite temperature 136
-1
0
1
2
3
4
5
6
7
10-3
10-2
10-1
100
101
102
O
S
(a)
-1
0
1
2
3
4
5
6
7
10-4
10-3
10-2
10-1
100
101
102
103
O
S
(b)
Figure 6.11: Scatter plots of S and O. (a) at T = 0, (b) at T = 0.95Tc for mps/Tc1.99.The blue dotted line closed with the y-axis corresponds to the region of good behavior.For simplicity the superscript label for the configuration is omitted in the plots.
respective measures to exclude the unphysical configurations. The subplots in Figure
6.11 are the scatter plots of the configurations on these above defined measures, S(i) and
O(i), for mπ/Tc = 1.99 studied at zero temperature and T = 0.95Tc. The blue dotted line
is the cut used in our calculations.
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
mps/T
zT
Oc = 01234
(a)
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
mps/T
zT
Sc = 58
1012
(b)
Figure 6.12: Effective mass plots for pseudoscalar correlators with varying cuts on S andO. The effective mass is observed to be more sensitive to O than S.
The effective mass (see eq. (3.2)) plots for the pseudoscalar, estimated for various
cuts attributed for the measures S(i) and O(i) are shown in Figure 6.12. It is evident
from the plots that the cut doesn’t affect the results much. In our calculations we took
configurations, which are well-behaved as those with ((S(i) ≤ 10) ∩ (O(i) ≤ 2))′. The (′)
Baryons at finite temperature 137
T/Tc κ N Nr
0 0.1347 100 710.95 0.1347 100 941.5 0.1355 70 67
Table 6.8: The total no. of configurations (N) generated and the no: of configurations(Nr) used for the analysis after discarding the configurations with strange behavior atdifferent temperatures.
is to emphasize that the cuts are applied only in those configurations with well-behaved
pseudoscalar correlator minima, as mentioned in the definition of the measure O. Table
6.8 contains the details regarding the no: of configurations generated and the no: of
configurations used as well-behaved configurations for the analysis.
6.B Wall sources
In this section, we make a detailed comparison of the screening correlator with the cor-
responding correlators in the FFT. We used wall sources for FFT, and compared the
effective masses, m(z), obtained on 322 × 8 × Nz lattices (with Nz = 80), with those
obtained in the interacting theory. Wall source at zero temperature is defined as the
collection of point sources with equal magnitude in all the points on a given time slice.
This is equivalent to a sum of the all the space indices at the source in a point-to-all
quark propagator. This can be expressed algebraically in terms of the point-to-all quark
propagator as
D−1w (t;x′, y′, z′, t′) =
∑x,y,z
D−1p (x, y, z, t;x′, y′, z′, t′), (6.19)
whereDp(x, y, z, t;x′, y′, z′, t′) is the point-to-all quark propagator, with source at (x, y, z, t)
and the sink at (x′, y′, z′, t′).
In Figure 6.13 we compare the PS and V screening masses for our lightest quarks
(corresponding to mπ/T = 1.52 at 1.5Tc) to screening masses in FFT with similar bare
quark mass. In the free theory, the screening correlator for wall source in these two
channels are equal. The results for the FFT were obtained on a lattice with Nz = 80
using the estimator lnC(z)/C(z + 1) for the screening masses. The plateaus in the
effective masses shown is due to the use of a wall source. When a point source is used,
neither the free theory nor the interacting theory shows a plateau [130]. These checks
give us confidence that we are able to extract the asymptotic behavior of the screening
Baryons at finite temperature 138
5.3
5.45
5.6
5.75
5.9
6.05
6.2
6.35
6.5
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
µP
S,V
s
/T
zT
PSVs
7.5
8
8.5
9
9.5
10
10.5
11
11.5
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
µN
-/T
zT
Figure 6.13: Effective masses, in units of temperature, for meson and nucleon sources, atT = 1.5Tc and in FFT. The black continuous curve denotes the FFT screening masses forNt = 8. The black dashed line shows the continuum value, µ = 2πT [129, 203]. The datapoints and the bands show the effective masses and the fitted estimate for the asymptoticvalue, respectively.
5.9
6.3
6.7
7.1
7.5
7.9
0.05 0.1 0.15 0.2 0.25 0.3 0.35
µP
S/T
Nt/Ns
8
9
10
11
12
13
0.05 0.1 0.15 0.2 0.25 0.3 0.35
µN
/T
Nt/Ns
Figure 6.14: Mesonic and nucleonic screening masses in FFT as a function of Nt/Ns =1/LT for Nt = 8. The filled symbols are estimates from wall source quark propagators,while the unfilled symbols are estimated using point sources. In the latter case, sincethere is no plateau in the local mass, we use the prescription of [130] and use the localmass at distance Nz/4.
correlator.
The effective masses for the negative parity nucleon are also shown in Figure 6.13.
In the interacting theory, a reasonable plateau is obtained in the effective mass, C(z) ∼exp(µ(z) × (Nz − z)). The plateau is more pronounced in our data for heavier quarks.
We see that the effective mass in the interacting theory is larger than that in the free
Baryons at finite temperature 139
theory.
Since we use Ns/Nt = LT = 4 we also check finite volume effects in a theory of free
quarks. In Figure 6.14 we show the screening masses for pseudoscalar meson and nucleon
on lattices with Nt = 8 and varying Ns. With wall sources we find essentially no finite
volume effect in either channels already for LT = 4. Since finite volume effects are larger
in the free theory than in the interacting theory, therefore finite volume effects in our
studies are expected to be small. In contrast, point sources give large finite volume effects
[130].
Chapter 7
Summary and conclusions
This thesis is a combination of two major works on baryons from lattice QCD. The first
part of it, involving charm baryon spectroscopy, is subdivided into two subparts : First
one dealing with the excited state spectroscopy of charm baryons, while the second work
is on charm hadron spectroscopy using overlap fermions. In the second part, we perform
a finite temperature study of screening masses of hadrons, particularly for baryons, where
we analyze the temperature dependence of the hadronic screening correlators above and
immediately below the deconfinement transition temperature.
7.1 Excited state charm baryon spectroscopy
This work constituting first segment of the first work covers major part of the thesis.
This work deals with the first non-perturbative calculation of the excited state spectra
of charm baryons with one, two and three charm quarks. In these calculations, we
extract highly excited states, which are very difficult to extract using the previously
existing lattice QCD techniques, of charm baryons. Utilizing the advanced lattice QCD
techniques like anisotropic lattice formalism, the derivative-based operator construction
and the variational fitting techniques, we extract the spectrum of all the charm baryons
having all possible flavor combinations with well-defined total spin up to 72. We also
perform several studies on various energy splittings between the charm hadrons. Being
first of its kind in the excited charm baryon spectra calculation, this study serves as a
foundation for several follow-up studies, which may help us to address many challenges
in the baryon spectroscopy. A summary of the results from this work is as follows.
We extract approximately 20 states for each of the charm baryons, along with a
140
Summary and conclusions 141
reliable identification of the spin-parity quantum numbers for each of these states. Beside
identifying the spin of a state we are also able to decode the structure of operators leading
to that state : whether constructed by relativistic, non-relativistic, hybrids, non-hybrid
types or a mixture of them all. Similar to light and strange baryon spectra [102], we also
find the number of extracted states of each spin in the three lowest-energy bands and the
number of quantum numbers expected based on weakly broken SU(6)×O(3) symmetry
agree perfectly, i.e., all the charm baryon spectra remarkably resemble the expectations
of quantum numbers from non-relativistic quark model [28, 29, 165]. Extracted spectra
do not show the signature of diquark spectra and the chiral symmetry restoration as
predicted by some models [204].
Since the spin dependent energy splittings can provide crucial information about
interaction energy we study various energy splittings, including splittings due to hyperfine
and spin-orbit coupling, for all the charm baryons. We find degeneracy between these
spin-orbit split states is more or less satisfied both for bottom and charm quarks. While
for the splittings, which mimics the binding energy of these states, comparison with the
respective splittings at other quark masses suggest that these splittings can modeled
with a form a+ b/mps derived from heavy quark effective theory. From the fitted results
involving both doubly and triply charm baryons, we are able to predict B∗c −Bc = 80± 8
MeV and Ω∗ccb(3/2+) = 8050 ± 10 MeV.
However, it is to be noted that pion mass used in this calculation is not physical and is
391 MeV. Furthermore, we have not used any multi-hadron operators in this calculation
which will play a major role at lighter pion masses. Inclusion of those operators, partic-
ularly those involving light quarks, may affect some of the above conclusions, though to
a lesser extent than their influence in the light hadron spectra. To get precise numbers
for the excited spectra one needs to follow up this calculation with physical pion mass
and multi-hadron operators.
The method used here to study charm baryons can also be employed for similar
study for bottom baryons. Ground state bottom baryons have been studied by lattice
QCD [80, 205, 206, 208, 207, 209]. However, so far excited state spectra from Lattice QCD
is limited only to a few low lying states [210] and triply bottom baryons (Ωbbb) [157] using
NRQCD. Considering experimental importance and prospects a comprehensive study,
using this method, is thus quite essential and an immediate extension of our work will
be carried out for bottom baryons in future.
Summary and conclusions 142
7.2 Charm hadron spectroscopy using overlap fermions
The second part of the spectroscopy work is precise non-perturbative calculation of the
low lying charm hadrons at multiple lattice spacings with improved control over the
cut off effects. Using a mixed action approach, we utilized the highly desirable features
of small discretization errors and small taste breaking effects of the HISQ action and
the small discretization effects and the low quark mass limit of the overlap fermion
action. We are thus able to study a long range of pseudoscalar meson masses from
230− 5000 MeV. By tuning the charm quark mass using the kinetic mass, we made sure
the discretization errors due to the uncertainty in the charm mass tuning is small. At this
tuned charm quark mass, we extracted the meson and the baryon spectra containing one
or more charm quarks. We made a comparative study between the results obtained in this
method to out previous study on excited charm baryons. Both results are also compared
with other lattice calculations and quark model calculations and experimental results,
where available. The consistency within the lattice calculations and the agreement with
the experimental values validates the credibility in both of our calculations. Thus, this
calculation serves simultaneously as a complementary study to compare the results from
our previous calculations and as an independent study with improved control over the
systematics that gives predictions about the low lying charm hadron spectrum.
It is also worthwhile to consider excited state study using overlap fermions. However,
the distillation method used for charm baryons will be prohibitively costly with extended
baryons. One may consider excited state calculations using only local operators but using
same flavor structures as those used in our study. The extension of overlap formulation
to bottom sector in our current formulation is not possible. However, one may think
about to use a full Fermilab type formulation to check whether it is at all possible to go
to bottom quark mass with minimum discretization error.
7.3 Nucleons at finite temperature
In the second work, we investigate the nature of the strongly interacting matter at high
temperatures using baryonic probes. In this work, we focused on the hadronic correla-
tion lengths that exist in a pure SU(3) gauge theory above and immediately below the
deconfinement transition temperature, Tc. As seen in the other lattice calculations, we
observed no statistically significant thermal effects in the mesonic correlations immedi-
ately below Tc. We also make the observation of the unphysical quenching artefacts in the
Summary and conclusions 143
scalar screening correlators immediately below Tc and make a first attempt to understand
the thermal effects on the ghost coupling in these correlators. We found no statistically
significant deviations with the available statistics in these studies also. Above Tc, clear
evidence of the chiral symmetry restoration in terms of the pairwise degeneracy in the
parity partner correlators and the weakly interacting nature of quarks was observed in
the mesonic channels.
Above Tc, baryonic correlations also show clear evidence of the chiral symmetry
restoration in terms of the pairwise degeneracy in the parity partner correlators and
propagate as three weakly interacting quarks. Below Tc, we find interesting behavior in
the nucleonic correlations, indicating precursor effects for the chiral symmetry restoration
in the nucleonic channels immediately below Tc. While no such effects have been seen
earlier either in the glue sector or with quarks, these observations are expected to have
implications for the analysis of the heavy ion collision data. These implications clearly
call for follow-up studies of nucleon around Tc. In future we like to do a comprehensive
calculation of baryonic correlations, including nucleonic, at various temperatures below
Tc and just above Tc.
Bibliography
[1] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).
[2] A. Salam, Conf. Proc. C 680519, 367 (1968).
[3] S. L. Glashow, Nucl. Phys. 22, 579 (1961).
[4] P. W. Higgs, Phys. Lett. 12, 132 (1964).
[5] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716, 30 (2012)
[arXiv:1207.7235 [hep-ex]].
[6] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716, 1 (2012) [arXiv:1207.7214
[hep-ex]].
[7] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973).
[8] H. D. Politzer, Phys. Rept. 14, 129 (1974).
[9] E. Klempt and J. -M. Richard, Rev. Mod. Phys. 82, 1095 (2010) and the references
therein.
[10] M. F. M. Lutz et al. [PANDA Collaboration], arXiv:0903.3905 [hep-ex].
[11] K. T. Brinkmann, P. Gianotti and I. Lehmann, Nucl. Phys. News 16, 15 (2006)
[physics/0701090].
[12] F. Wick [CDF Collaboration], Int. J. Mod. Phys. Conf. Ser. 02, 163 (2011)
[arXiv:1105.0517 [hep-ex]].
[13] RAaij et al. [LHCb Collaboration], JHEP 1312, 090 (2013) [arXiv:1310.2538 [hep-
ex]].
[14] T. Abe et al. [Belle-II Collaboration], arXiv:1011.0352 [physics.ins-det].
144
Bibliography 145
[15] D. M. Asner, T. Barnes, J. M. Bian, I. I. Bigi, N. Brambilla, I. R. Boyko, V. Bytev
and K. T. Chao et al., Int. J. Mod. Phys. A 24, S1 (2009) [arXiv:0809.1869 [hep-ex]].
[16] RPP : J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012).
[17] M. Mattson et al. [SELEX Collaboration], Phys. Rev. Lett. 89, 112001 (2002) [hep-
ex/0208014].
[18] J. S. Russ [SELEX Collaboration], hep-ex/0209075.
[19] A. Ocherashvili et al. [SELEX Collaboration], Phys. Lett. B 628, 18 (2005) [hep-
ex/0406033].
[20] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 74, 011103 (2006) [hep-
ex/0605075].
[21] Y. Kato et al. [Belle Collaboration], arXiv:1312.1026 [hep-ex].
[22] R. Chistov et al. [BELLE Collaboration], Phys. Rev. Lett. 97, 162001 (2006) [hep-
ex/0606051].
[23] S. Capstick and W. Roberts, Prog. Part. Nucl. Phys. 45, S241 (2000) and the refer-
ences therein.
[24] V. Crede and W. Roberts, Rept. Prog. Phys. 76, 076301 (2013) and the references
therein.
[25] J. D. Bjorken, FERMILAB-CONF-85/69, Is the ccc a New Deal for Baryon Spec-
troscopy?, Int. Conf. on Hadron Spectroscopy, College Park, MD, Apr. 1985;
[26] A. Pineda and J. Soto, Nucl. Phys. Proc. Suppl. 64, 428 (1998) [hep-ph/9707481].
[27] N. Brambilla, A. Pineda, J. Soto and A. Vairo, Nucl. Phys. B 566, 275 (2000)
[hep-ph/9907240].
[28] N. Isgur and G. Karl, Phys. Lett. B 72, 109 (1977).
[29] N. Isgur and G. Karl, Phys. Rev. D 18, 4187 (1978).
[30] P. Hasenfratz, R. R. Horgan, J. Kuti and J. M. Richard, Phys. Lett. B 94, 401
(1980).
Bibliography 146
[31] Y. Jia, JHEP 0610, 073 (2006) [hep-ph/0607290].
[32] J. Vijande, H. Garcilazo, A. Valcarce and F. Fernandez, Phys. Rev. D 70, 054022
(2004) [hep-ph/0408274].
[33] H. Garcilazo, J. Vijande and A. Valcarce, J. Phys. G 34, 961 (2007) [hep-
ph/0703257].
[34] B. Silvestre-Brac, Few Body Syst. 20, 1 (1996).
[35] J. G. Korner, M. Kramer and D. Pirjol, Prog. Part. Nucl. Phys. 33, 787 (1994)
[hep-ph/9406359].
[36] C. Albertus, J. E. Amaro, E. Hernandez and J. Nieves, Nucl. Phys. A 740, 333
(2004) [nucl-th/0311100].
[37] C. Albertus, E. Hernandez, J. Nieves and J. M. Verde-Velasco, Eur. Phys. J. A 31,
691 (2007) [hep-ph/0610131].
[38] J. M. Flynn, E. Hernandez and J. Nieves, Phys. Rev. D 85, 014012 (2012)
[arXiv:1110.2962 [hep-ph]].
[39] E. E. Jenkins, Phys. Rev. D 54 (1996) 4515 [hep-ph/9603449].
[40] W. Roberts and M. Pervin, Int. J. Mod. Phys. A 23, 2817 (2008) [arXiv:0711.2492
[nucl-th]].
[41] N. Brambilla, A. Pineda, J. Soto and A. Vairo, Rev. Mod. Phys. 77, 1423 (2005)
[hep-ph/0410047].
[42] N. Brambilla, A. Vairo and T. Rosch, Phys. Rev. D 72, 034021 (2005) [hep-
ph/0506065].
[43] N. Brambilla, J. Ghiglieri and A. Vairo, Phys. Rev. D 81, 054031 (2010)
[arXiv:0911.3541 [hep-ph]].
[44] F. J. Llanes-Estrada, O. I. Pavlova and R. Williams, Eur. Phys. J. C 72, 2019 (2012)
[arXiv:1111.7087 [hep-ph]].
[45] B. A. Thacker and G. P. Lepage, Phys. Rev. D 43, 196 (1991).
Bibliography 147
[46] G. P. Lepage, L. Magnea, C. Nakhleh, U. Magnea and K. Hornbostel, Phys. Rev. D
46, 4052 (1992) [hep-lat/9205007].
[47] H. D. Trottier, Phys. Rev. D 55, 6844 (1997) [hep-lat/9611026].
[48] C. T. H. Davies et al. [UKQCD Collaboration], Phys. Rev. D 58, 054505 (1998)
[hep-lat/9802024].
[49] S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986).
[50] A. P. Martynenko, Phys. Lett. B 663, 317 (2008) [arXiv:0708.2033 [hep-ph]].
[51] S. Migura, D. Merten, B. Metsch and H. -R. Petry, Eur. Phys. J. A 28, 41 (2006)
[hep-ph/0602153].
[52] S. M. Gerasyuta and E. E. Matskevich, Int. J. Mod. Phys. E 17, 585 (2008)
[arXiv:0709.0397 [hep-ph]].
[53] D. Ebert, R. N. Faustov and V. O. Galkin, Phys. Lett. B 659, 612 (2008)
[arXiv:0705.2957 [hep-ph]].
[54] J. -R. Zhang and M. -Q. Huang, Phys. Lett. B 674, 28 (2009) [arXiv:0902.3297
[hep-ph]].
[55] Z. -G. Wang, Commun. Theor. Phys. 58, 723 (2012) [arXiv:1112.2274 [hep-ph]].
[56] X. -H. Guo, K. -W. Wei and X. -H. Wu, Phys. Rev. D 78, 056005 (2008)
[arXiv:0809.1702 [hep-ph]].
[57] F. Butler, H. Chen, J. Sexton, A. Vaccarino and D. Weingarten, Nucl. Phys. B 430,
179 (1994) [hep-lat/9405003].
[58] S. Aoki, et al. [CP-PACS Collaboration], Phys. Rev. Lett. 84, 238 (2000) [hep-
lat/9904012]
[59] C. W. Bernard, et al., Phys. Rev. D 64, 054506 (2001) [hep-lat/0104002].
[60] C. Aubin, et al., Phys. Rev. D 70, 094505 (2004) [hep-lat/0402030].
[61] S. Aoki et al. [PACS-CS Collaboration], Phys. Rev. D 79, 034503 (2009)
[arXiv:0807.1661 [hep-lat]].
Bibliography 148
[62] D. J. Antonio et al. [RBC and UKQCD Collaborations], Phys. Rev. D 75, 114501
(2007) [hep-lat/0612005].
[63] A. Walker-Loud, et al., Phys. Rev. D 79, 054502 (2009) [arXiv:0806.4549 [hep-lat]].
[64] C. Alexandrou et al. [European Twisted Mass Collaboration], Phys. Rev. D 78,
014509 (2008) [arXiv:0803.3190 [hep-lat]].
[65] S. Durr, et al., Science 322, 1224 (2008) [arXiv:0906.3599 [hep-lat]].
[66] C. T. H. Davies, K. Hornbostel, A. Langnau, G. P. Lepage, A. Lidsey, J. Shigemitsu
and J. H. Sloan, Phys. Rev. D 50, 6963 (1994) [hep-lat/9406017].
[67] C. T. H. Davies, K. Hornbostel, G. P. Lepage, A. J. Lidsey, J. Shigemitsu and
J. H. Sloan, Phys. Rev. D 52, 6519 (1995) [hep-lat/9506026].
[68] M. Okamoto et al. [CP-PACS Collaboration], Phys. Rev. D 65, 094508 (2002) [hep-
lat/0112020].
[69] A. Gray, I. Allison, C. T. H. Davies, E. Dalgic, G. P. Lepage, J. Shigemitsu and
M. Wingate, Phys. Rev. D 72, 094507 (2005) [hep-lat/0507013].
[70] C. Aubin, C. Bernard, C. E. DeTar, M. Di Pierro, E. D. Freeland, S. Gottlieb,
U. M. Heller and J. E. Hetrick et al., Phys. Rev. Lett. 95, 122002 (2005) [hep-
lat/0506030].
[71] A. Bazavov et al. [Fermilab Lattice and MILC Collaborations], Phys. Rev. D 85,
114506 (2012) [arXiv:1112.3051 [hep-lat]].
[72] T. Burch, C. DeTar, M. Di Pierro, A. X. El-Khadra, E. D. Freeland, S. Got-
tlieb, A. S. Kronfeld and L. Levkova et al., Phys. Rev. D 81, 034508 (2010)
[arXiv:0912.2701 [hep-lat]].
[73] G. C. Donald, C. T. H. Davies, R. J. Dowdall, E. Follana, K. Hornbostel, J. Koponen,
G. P. Lepage and C. McNeile, Phys. Rev. D 86, 094501 (2012) [arXiv:1208.2855
[hep-lat]].
[74] C. McNeile, et al., Phys. Rev. D 86, 074503 (2012) [arXiv:1207.0994 [hep-lat]].
[75] E. B. Gregory, et al., Phys. Rev. D 83, 014506 (2011) [arXiv:1010.3848 [hep-lat]].
Bibliography 149
[76] R. J. Dowdall et al. [HPQCD Collaboration], Phys. Rev. D 85, 054509 (2012)
[arXiv:1110.6887 [hep-lat]].
[77] R. J. Dowdall, C. T. H. Davies, T. C. Hammant and R. R. Horgan, Phys. Rev. D
86, 094510 (2012) [arXiv:1207.5149 [hep-lat]].
[78] Y. Namekawa et al. [PACS-CS Collaboration], Phys. Rev. D 84, 074505 (2011)
[arXiv:1104.4600 [hep-lat]].
[79] D. Mohler and R. M. Woloshyn, Phys. Rev. D 84, 054505 (2011) [arXiv:1103.5506
[hep-lat]];
[80] N. Mathur, R. Lewis and R. M. Woloshyn, Phys. Rev. D 66, 014502 (2002) [hep-
ph/0203253]
[81] R. Lewis, N. Mathur and R. M. Woloshyn, Phys. Rev. D 64, 094509 (2001) [hep-
ph/0107037].
[82] J. M. Flynn, et al., [UKQCD Collaboration], JHEP 0307, 066 (2003) [hep-
lat/0307025].
[83] L. Liu, et al., Phys. Rev. D 81, 094505 (2010) [arXiv:0909.3294 [hep-lat]].
[84] R. A. Briceno, H. -W. Lin and D. R. Bolton, Phys. Rev. D 86, 094504 (2012)
[arXiv:1207.3536 [hep-lat]].
[85] C. Alexandrou, J. Carbonell, D. Christaras, V. Drach, M. Gravina and M. Papinutto,
Phys. Rev. D 86, 114501 (2012) [arXiv:1205.6856 [hep-lat]].
[86] G. Bali, et al., J. Phys. Conf. Ser. 426, 012017 (2013) [arXiv:1212.0565 [hep-lat]].
[87] S. Durr, G. Koutsou and T. Lippert, Phys. Rev. D 86, 114514 (2012)
[arXiv:1208.6270 [hep-lat]].
[88] Y. Namekawa et al. [PACS-CS Collaboration], Phys. Rev. D 87, 094512 (2013)
[arXiv:1301.4743 [hep-lat]].
[89] N. Mathur, Y. Chen, S. J. Dong, T. Draper, I. Horvath, F. X. Lee, K. F. Liu and
J. B. Zhang, Phys. Lett. B 605, 137 (2005) [hep-ph/0306199].
[90] D. B. Leinweber, W. Melnitchouk, D. G. Richards, A. G. Williams and J. M. Zanotti,
Lect. Notes Phys. 663, 71 (2005) [nucl-th/0406032].
Bibliography 150
[91] K. Sasaki, S. Sasaki and T. Hatsuda, Phys. Lett. B 623, 208 (2005) [hep-
lat/0504020].
[92] T. Burch, C. Gattringer, L. Y. .Glozman, C. Hagen, D. Hierl, C. B. Lang and
A. Schafer, Phys. Rev. D 74, 014504 (2006) [hep-lat/0604019].
[93] M. S. Mahbub et al. [CSSM Lattice Collaboration], Phys. Lett. B 707, 389 (2012)
[arXiv:1011.5724 [hep-lat]].
[94] M. S. Mahbub, A. O. Cais, W. Kamleh, D. B. Leinweber and A. G. Williams, Phys.
Rev. D 82, 094504 (2010) [arXiv:1004.5455 [hep-lat]].
[95] G. P. Engel, C. B. Lang and A. Schafer, Phys. Rev. D 87, no. 3, 034502 (2013)
[arXiv:1212.2032 [hep-lat]].
[96] M. Peardon, et al., [Hadron Spectrum Collaboration], Phys. Rev. D 80, 054506
(2009) [arXiv:0905.2160 [hep-lat]].
[97] C. Michael, Nucl. Phys. B 259, 58 (1985);
[98] M. Luscher and U. Wolff, Nucl. Phys. B 339, 222 (1990).
[99] J. J. Dudek, R. G. Edwards, N. Mathur and D. G. Richards, Phys. Rev. D 77,
034501 (2008) [arXiv:0707.4162 [hep-lat]].
[100] S. Basak et al. [Lattice Hadron Physics (LHPC) Collaboration], Phys. Rev. D 72,
074501 (2005) [hep-lat/0508018].
[101] R. G. Edwards, J. J. Dudek, D. G. Richards and S. J. Wallace, Phys. Rev. D 84,
074508 (2011) [arXiv:1104.5152 [hep-ph]].
[102] R. G. Edwards, N. Mathur, D. G. Richards and S. J. Wallace, Phys. Rev. D 87,
054506 (2013) arXiv:1212.5236 [hep-ph].
[103] J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards and C. E. Thomas,
Phys. Rev. Lett. 103, 262001 (2009), [arXiv:0909.0200 [hep-ph]]
[104] J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards and C. E. Thomas,
Phys. Rev. D 82, 034508 (2010) [arXiv:1004.4930 [hep-ph]].
[105] J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards and C. E. Thomas,
Phys. Rev. D 83, 071504 (2011) [arXiv:1011.6352 [hep-ph]].
Bibliography 151
[106] L. Liu, et al., [HSC], JHEP 1207, 126 (2012) [arXiv:1204.5425 [hep-ph]].
[107] G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas and L. Liu, JHEP 1305, 021
(2013) [arXiv:1301.7670 [hep-ph]].
[108] J. J. Dudek and R. G. Edwards, Phys. Rev. D 85, 054016 (2012) [arXiv:1201.2349
[hep-ph]].
[109] R. G. Edwards, B. Joo and H. -W. Lin, Phys. Rev. D 78, 054501 (2008)
[arXiv:0803.3960 [hep-lat]].
[110] H. -W. Lin et al. [HSC], Phys. Rev. D 79, 034502 (2009) [arXiv:0810.3588 [hep-lat]].
[111] G. P. Lepage and P. B. Mackenzie, Phys. Rev. D 48, 2250 (1993) [hep-lat/9209022].
[112] M. Padmanath, R. G. Edwards, N. Mathur and M. Peardon, arXiv:1307.7022 [hep-
lat].
[113] M. Padmanath, R. G. Edwards, N. Mathur and M. Peardon, arXiv:1311.4806 [hep-
lat].
[114] M. Padmanath, R. G. Edwards, N. Mathur and M. Peardon, arXiv:1311.4354 [hep-
lat].
[115] D. B. Renner et al. [LHP Collaboration], Nucl. Phys. Proc. Suppl. 140, 255 (2005)
[hep-lat/0409130].
[116] R. G. Edwards et al. [LHPC Collaboration], Phys. Rev. Lett. 96, 052001 (2006)
[hep-lat/0510062].
[117] J. D. Bratt et al. [LHPC Collaboration], Phys. Rev. D 82, 094502 (2010)
[arXiv:1001.3620 [hep-lat]].
[118] C. Allton, C. Maynard, A. Trivini and R. Tweedie, PoS LAT 2006, 202 (2006)
[hep-lat/0610068].
[119] S. Durr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, T. .Kurth, L. Lellouch and
T. .Lippert et al., PoS LAT 2007, 113 (2007) [arXiv:0710.4866 [hep-lat]].
[120] K. Cichy, G. Herdoiza and K. Jansen, Acta Phys. Polon. Supp. 2, 497 (2009)
[arXiv:0910.0816 [hep-lat]].
Bibliography 152
[121] E. Follana et al. [HPQCD and UKQCD Collaborations], Phys. Rev. D 75, 054502
(2007) [hep-lat/0610092].
[122] A. Bazavov et al. [MILC Collaboration], Phys. Rev. D 87, no. 5, 054505 (2013)
[arXiv:1212.4768 [hep-lat]].
[123] R. Narayanan and H. Neuberger, Phys. Rev. Lett. 71, 3251 (1993) [hep-
lat/9308011].
[124] H. Neuberger, Phys. Lett. B 417, 141 (1998); ; ibid. B427 (1998) 353;
[125] R. G. Edwards, U. M. Heller and R. Narayanan, Phys. Rev. D 59, 094510 (1999)
[hep-lat/9811030].
[126] S. Basak, et al., [ILGTI] PoS LATTICE 2012, 141 (2012) [arXiv:1211.6277 [hep-
lat]].
[127] S. Basak, S. Datta, M. Padmanath, P. Majumdar and N. Mathur, PoS LATTICE
2012, 141 (2012) [arXiv:1211.6277 [hep-lat]].
[128] C. E. DeTar and J. B. Kogut, Phys. Rev. Lett. 59 (1987) 399, Phys. Rev. D 36
(1987) 2828.
[129] K. D. Born et al.(MTc Collaboration), Phys. Rev. Lett. 67 (1991) 302.
[130] S. Wissel et al., PoS LAT-2005, 164; S. Wissel, Ph. D dissertation (2006).
[131] R. V. Gavai, S. Gupta and R. Lacaze, Phys. Rev. D 78 (2008) 014502.
[132] R. V. Gavai, S. Gupta and P. Majumdar, Phys. Rev. D 65 (2002) 054506.
[133] A. Gocksch, P. Rossi and Urs M. Heller, Phys. Lett. B 205 (1988) 334.
[134] M. Cheng et al., Eur. Phys. J. C 71 (2011) 1564;
D. Banerjee et al., Phys. Rev. D 83 (2011) 074510.
[135] M. Cheng et al., Nucl. Phys. A 862–863 (2011) 308.
[136] S. Datta, S. Gupta, M. Padmanath, J. Maiti and N. Mathur, JHEP 1302, 145
(2013) [arXiv:1212.2927 [hep-lat]].
Bibliography 153
[137] K. G. Wilson, Phys. Rev. D 10, 2445 (1974).
[138] T. DeGrand and C. E. Detar, New Jersey, USA: World Scientific (2006) 345 p
[139] C. Gattringer and C. B. Lang, Lect. Notes Phys. 788, 1 (2010).
[140] I. Montvay and G. Munster, Cambridge, UK: Univ. Pr. (1994) 491 p. (Cambridge
monographs on mathematical physics)
[141] P. T. Matthews and A. Salam, Nuovo Cim. 2, 120 (1955).
[142] H. B. Nielsen and M. Ninomiya, Phys. Lett. B 105, 219 (1981).
[143] B. Sheikholeslami and R. Wohlert, Nucl. Phys. B 259, 572 (1985).
[144] M. Luscher, S. Sint, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys. B 491, 323
(1997) [hep-lat/9609035].
[145] J. B. Kogut and L. Susskind, Phys. Rev. D 11, 395 (1975).
[146] S. R. Sharpe, PoS LAT 2006, 022 (2006) [hep-lat/0610094].
[147] M. Luscher, Phys. Lett. B 428, 342 (1998) [hep-lat/9802011].
[148] M. A. Clark and A. D. Kennedy, Nucl. Phys. Proc. Suppl. 129, 850 (2004) [hep-
lat/0309084].
[149] M. A. Clark, A. D. Kennedy and Z. Sroczynski, Nucl. Phys. Proc. Suppl. 140, 835
(2005) [hep-lat/0409133].
[150] M. A. Clark and A. D. Kennedy, Phys. Rev. Lett. 98, 051601 (2007) [hep-
lat/0608015].
[151] G. P. Lepage, CLNS-89-971.
[152] F. Karsch, Nucl. Phys. B 205, 285 (1982).
[153] G. Burgers, F. Karsch, A. Nakamura and I. O. Stamatescu, Nucl. Phys. B 304,
587 (1988);
[154] T. R. Klassen, Nucl. Phys. B 533, 557 (1998); [hep-lat/9803010].
[155] P. Chen, Phys. Rev. D 64, 034509 (2001). [hep-lat/0006019].
Bibliography 154
[156] C. R. Allton et al. [UKQCD Collaboration], Phys. Rev. D 47, 5128 (1993) [hep-
lat/9303009].
[157] S. Meinel, Phys. Rev. D 85, 114510 (2012).
[158] N. Brambilla, S. Eidelman, B. K. Heltsley, R. Vogt, G. T. Bodwin, E. Eichten,
A. D. Frawley and A. B. Meyer et al., Eur. Phys. J. C 71, 1534 (2011)
[arXiv:1010.5827 [hep-ph]].
[159] M. J. Savage and M. B. Wise, Phys. Lett. B 248, 177 (1990).
[160] M. J. Savage and R. P. Springer, Int. J. Mod. Phys. A 6, 1701 (1991).
[161] S. Fleming and T. Mehen, Phys. Rev. D 73, 034502 (2006) [hep-ph/0509313].
[162] E. J. Eichten and C. Quigg, Phys. Rev. D 49, 5845 (1994) [hep-ph/9402210].
[163] E. B. Gregory, C. T. H. Davies, E. Follana, E. Gamiz, I. D. Kendall, G. P. Lepage,
H. Na and J. Shigemitsu et al., Phys. Rev. Lett. 104, 022001 (2010) [arXiv:0909.4462
[hep-lat]].
[164] Z. S. Brown, W. Detmold, S. Meinel and K. Orginos, PoS QNP 2012, 107 (2012).
[165] O. W. Greenberg, Phys. Rev. Lett. 13, 598 (1964).
[166] J. J. Dudek, R. G. Edwards and C. E. Thomas, Phys. Rev. D 87, no. 3, 034505
(2013) [arXiv:1212.0830 [hep-ph]].
[167] H. Fukaya et al. [JLQCD Collaboration], Phys. Rev. Lett. 98, 172001 (2007) [hep-
lat/0702003].
[168] A. Li et al. [χQCD Collaboration], Phys. Rev. D 82, 114501 (2010) [arXiv:1005.5424
[hep-lat]].
[169] O. Bar, G. Rupak and N. Shoresh, Phys. Rev. D 67, 114505 (2003) [hep-
lat/0210050].
[170] J. -W. Chen, D. O’Connell and A. Walker-Loud, Phys. Rev. D 75, 054501 (2007)
[hep-lat/0611003].
[171] K. Orginos and A. Walker-Loud, Phys. Rev. D 77, 094505 (2008) [arXiv:0705.0572
[hep-lat]].
Bibliography 155
[172] N. Mathur et al. [xQCD Collaboration], PoS LATTICE 2010, 114 (2010)
[arXiv:1011.4378 [hep-lat]].
[173] S. Capitani, S. Durr and C. Hoelbling, JHEP 0611, 028 (2006) [hep-lat/0607006].
[174] J. van den Eshof, A. Frommer, T. Lippert, K. Schilling and H. A. van der Vorst,
Comput. Phys. Commun. 146, 203 (2002) [hep-lat/0202025].
[175] Y. Chen, S. J. Dong, T. Draper, I. Horvath, F. X. Lee, K. F. Liu, N. Mathur and
J. B. Zhang, Phys. Rev. D 70, 034502 (2004) [hep-lat/0304005].
[176] C. T. H. Davies et al. [HPQCD Collaboration], Phys. Rev. D 81, 034506 (2010)
[arXiv:0910.1229 [hep-lat]].
[177] S. Tamhankar, A. Alexandru, Y. Chen, S. J. Dong, T. Draper, I. Horvath, F. X. Lee
and K. F. Liu et al., Phys. Lett. B 638, 55 (2006) [hep-lat/0507027].
[178] A. X. El-Khadra, A. S. Kronfeld and P. B. Mackenzie, Phys. Rev. D 55, 3933
(1997) [hep-lat/9604004].
[179] M. Benmerrouche, R. M. Davidson and N. C. Mukhopadhyay, Phys. Rev. C 39,
2339 (1989).
[180] S. Gupta, Phys. Rev. D 64 (2001) 034507.
[181] G. Boyd et al., Nucl. Phys. B 469 (1996) 419.
[182] D. Becirevic et al., Phys. Lett. B 558 (2003) 69.
[183] W. Bardeen et al., Phys. Rev. D 65 (2001) 014509.
[184] N. Mathur et al., Phys. Rev. D 76 (2007) 114505.
[185] S. Prelovsek et al., Phys. Rev. D 70 (2004) 094503.
[186] M. Luscher et al., Nucl. Phys. B 491 (1997) 344.
[187] G. P. Lepage and P. Mackenzie, Phys. Rev. D 48 (1993) 2250.
[188] M. Gockeler et al., Nucl. Phys. B (Proc. Suppl.) 53 (1997) 896.
[189] S. Sint and P. Weisz, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 856.
Bibliography 156
[190] S. Datta and S. Gupta, Phys. Rev. D 80 (2009) 114504.
[191] S. Gupta, Phys. Rev. D 60 (1999) 094505.
[192] R. Rapp and J. Wambach, Adv. Nucl. Phys. 25 (2000) 1.
[193] S. Gupta and N. Karthik, Phys. Rev. D 87, 094001 (2013) [arXiv:1302.4917 [hep-
lat]].
[194] M. Luscher, S. Sint, R. Sommer, P. Weisz, U. Wolff, Nucl. Phys. B 491 (1991),
323− 343.
[195] T. DeGrand, A. Hasenfratz and T. G. Kovacs, Nucl. Phys. B 547 (1999) 259−280.
[196] W. Bardeen, A. Duncan, E. Eichten, G. Hockney and H. Thacker, Phys. Rev. D
57, 1633− 1641, (1998).
[197] M. Gockeler, A. Hoferichter, R. Horsley, D. Pleiter, P. Rakow, G. Schierholz and
P. Stephenson, Nucl. Phys. B (Proc. Suppl.) 73 (1999) 889− 891.
[198] A. Hoferichter, E. Laermann, V. K. Mitrjushkin, M. Muller-Preussker and P.
Schmidt, Nucl. Phys. B (Proc. Suppl.) 63A-C (1998) 164− 166.
[199] R. Frezzotti, P. A. Grassi, S. Sint and P. Weisz, Nucl. Phys. B (Proc. Suppl.),
83− 84 (2000) 941− 946.
[200] C. Gattringer and S. Solbrig, PoS, LAT2005:127.
[201] C. Bernard, T. DeGrand, C. DeTar, S. Gottlieb, Urs M. Heller, C. McNeile, K.
Originos, R. Sugar and D. Toussaint, Nucl.Phys. B (Proc.Suppl.) 83 (2000) 289−291.
[202] C. Bernard, S. Datta, T. DeGrand, C. DeTar, S. Gottlieb, Urs M. Heller, C. Mc-
Neile, K. Originos, R. Sugar and D. Toussaint, Phys. Rev. D 66 (2002) 094501.
[203] V. I. Eletskii and B. L. Ioffe, Sov. J. Nucl. Phys. 48 (1988) 384;
W. Florkowski and B. L. Friman, Z. Phys. A 347 (1994) 271.
[204] L. Y. .Glozman, C. B. Lang and M. Schrock, Phys. Rev. D 86, 014507 (2012)
[arXiv:1205.4887 [hep-lat]].
[205] K. C. Bowler et al. [UKQCD Collaboration], Phys. Rev. D 54, 3619 (1996) [hep-
lat/9601022].
Bibliography 157
[206] A. Ali Khan, T. Bhattacharya, S. Collins, C. T. H. Davies, R. Gupta, C. Morn-
ingstar, J. Shigemitsu and J. H. Sloan, Phys. Rev. D 62, 054505 (2000) [hep-
lat/9912034].
[207] R. Lewis and R. M. Woloshyn, Phys. Rev. D 79, 014502 (2009) [arXiv:0806.4783
[hep-lat]].
[208] W. Detmold, C. -J. D. Lin and M. Wingate, Nucl. Phys. B 818, 17 (2009)
[arXiv:0812.2583 [hep-lat]].
[209] H. -W. Lin, S. D. Cohen, N. Mathur and K. Orginos, Phys. Rev. D 80, 054027
(2009) [arXiv:0905.4120 [hep-lat]].
[210] T. Burch, C. Hagen, C. B. Lang, M. Limmer and A. Schafer, Phys. Rev. D 79,
014504 (2009) [arXiv:0809.1103 [hep-lat]].
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